Datasets:

l3lab

/

ntp-mathlib

like 2

Follow

L3 Lab 3

case hg.hf x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x ⊢ ConvexOn ℝ (Set.Ioi 0.5) fun x => x

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1)

exact convexOn_id (convex_Ioi (0.5 : ℝ))

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1)

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f ⊢ f x ≤ 0

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ))

suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ))

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 ⊢ f x ≤ 0

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by

obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

case intro.intro.intro.intro.intro.intro x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f x1 x2 : ℝ h1 : 0.5 < x1 h2 : x1 < x2 h0 : x2 ≤ x h3 : 0 ≤ f x1 h4 : f x2 ≤ 0 ⊢ f x ≤ 0

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this

exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f ⊢ ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4

refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f ⊢ 0.5 < 18

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by

norm_num1

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f ⊢ 18 < 512

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by

norm_num1

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

case refine'_1 x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f ⊢ 0 ≤ f 18

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ ·

have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ ·

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f ⊢ 0 < 6

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by

norm_num1

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f ⊢ 6 * 6 = 2 * 18

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by

norm_num1

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

case refine'_1 x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 18) = 6 ⊢ 0 ≤ f 18

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)

rw [hf, log_nonneg_iff, this]

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

case refine'_1 x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 18) = 6 ⊢ 1 ≤ 18 * (2 * 18) ^ 6 / 4 ^ (18 / 3)

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this]

rw [one_le_div]

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this]

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

case refine'_1 x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 18) = 6 ⊢ 4 ^ (18 / 3) ≤ 18 * (2 * 18) ^ 6

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;>

norm_num1

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;>

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

case refine'_1 x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 18) = 6 ⊢ 0 < 4 ^ (18 / 3)

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;>

norm_num1

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;>

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

case refine'_1 x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 18) = 6 ⊢ 4 ^ 6 ≤ 18 * 36 ^ 6

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1

apply le_trans _ (le_mul_of_one_le_left _ _)

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 18) = 6 ⊢ 4 ^ 6 ≤ 36 ^ 6

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;>

norm_num1

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;>

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 18) = 6 ⊢ 0 ≤ 36 ^ 6

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;>

norm_num1

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;>

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 18) = 6 ⊢ 1 ≤ 18

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;>

norm_num1

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;>

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 18) = 6 ⊢ 4 ^ 6 ≤ 36 ^ 6

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1

apply Real.rpow_le_rpow

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

case h x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 18) = 6 ⊢ 0 ≤ 4

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;>

norm_num1

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;>

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

case h₁ x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 18) = 6 ⊢ 4 ≤ 36

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;>

norm_num1

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;>

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

case h₂ x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 18) = 6 ⊢ 0 ≤ 6

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;>

norm_num1

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;>

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 18) = 6 ⊢ 0 ≤ 36 ^ 6 case refine'_1 x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 18) = 6 ⊢ 0 < 4 ^ 6 case refine'_1 x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 18) = 6 ⊢ 0 < 18 * (2 * 18) ^ sqrt (2 * 18) / 4 ^ (18 / 3) case refine'_1.a x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 18) = 6 ⊢ 0 < 18

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1

apply rpow_nonneg_of_nonneg

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

case hx x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 18) = 6 ⊢ 0 ≤ 36 case refine'_1 x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 18) = 6 ⊢ 0 < 4 ^ 6 case refine'_1 x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 18) = 6 ⊢ 0 < 18 * (2 * 18) ^ sqrt (2 * 18) / 4 ^ (18 / 3) case refine'_1.a x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 18) = 6 ⊢ 0 < 18

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg;

norm_num1

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg;

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

case refine'_1 x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 18) = 6 ⊢ 0 < 4 ^ 6 case refine'_1 x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 18) = 6 ⊢ 0 < 18 * (2 * 18) ^ sqrt (2 * 18) / 4 ^ (18 / 3) case refine'_1.a x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 18) = 6 ⊢ 0 < 18

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1

apply rpow_pos_of_pos

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

case refine'_1.hx x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 18) = 6 ⊢ 0 < 4 case refine'_1 x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 18) = 6 ⊢ 0 < 18 * (2 * 18) ^ sqrt (2 * 18) / 4 ^ (18 / 3) case refine'_1.a x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 18) = 6 ⊢ 0 < 18

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos;

norm_num1

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos;

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

case refine'_1 x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 18) = 6 ⊢ 0 < 18 * (2 * 18) ^ sqrt (2 * 18) / 4 ^ (18 / 3) case refine'_1.a x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 18) = 6 ⊢ 0 < 18

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1

apply hf' 18

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

case refine'_1 x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 18) = 6 ⊢ 0 < 18 case refine'_1.a x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 18) = 6 ⊢ 0 < 18

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18;

norm_num1

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18;

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

case refine'_1.a x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 18) = 6 ⊢ 0 < 18

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1

norm_num1

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

case refine'_2 x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f ⊢ f 512 ≤ 0

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 ·

have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 ·

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f ⊢ 0 < 32

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by

norm_num1

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f ⊢ 32 * 32 = 2 * 512

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by

norm_num1

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

case refine'_2 x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 512) = 32 ⊢ f 512 ≤ 0

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)

rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one]

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

case refine'_2 x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 512) = 32 ⊢ 512 * (2 * 512) ^ 32 ≤ 4 ^ (512 / 3)

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;>

norm_num1

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;>

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

case refine'_2 x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 512) = 32 ⊢ 0 < 4 ^ (512 / 3)

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;>

norm_num1

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;>

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 512) = 32 ⊢ 0 < 512

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;>

norm_num1

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;>

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

case refine'_2.a x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 512) = 32 ⊢ 0 < 512

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;>

norm_num1

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;>

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

case refine'_2 x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 512) = 32 ⊢ 512 * 1024 ^ 32 ≤ 4 ^ (512 / 3)

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 ·

conv in 512 => equals 2 ^ 9 => norm_num1

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 ·

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 512) = 32 | 512

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 =>

equals 2 ^ 9 => norm_num1

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 =>

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 512) = 32 | 512

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 =>

equals 2 ^ 9 => norm_num1

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 =>

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 512) = 32 | 512

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 =>

equals 2 ^ 9 => norm_num1

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 =>

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 512) = 32 ⊢ 512 = 2 ^ 9

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 =>

norm_num1

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 =>

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

case refine'_2 x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 512) = 32 ⊢ 2 ^ 9 * 1024 ^ 32 ≤ 4 ^ (512 / 3)

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1

conv in 1024 => equals 2 ^ 10 => norm_num1

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 512) = 32 | 1024

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 =>

equals 2 ^ 10 => norm_num1

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 =>

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 512) = 32 | 1024

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 =>

equals 2 ^ 10 => norm_num1

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 =>

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 512) = 32 | 1024

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 =>

equals 2 ^ 10 => norm_num1

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 =>

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 512) = 32 ⊢ 1024 = 2 ^ 10

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 =>

norm_num1

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 =>

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

case refine'_2 x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 512) = 32 ⊢ 2 ^ 9 * (2 ^ 10) ^ 32 ≤ 4 ^ (512 / 3)

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1

conv in 32 => rw [← Nat.cast_ofNat]

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 512) = 32 | 32

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 =>

rw [← Nat.cast_ofNat]

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 =>

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 512) = 32 | 32

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 =>

rw [← Nat.cast_ofNat]

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 =>

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 512) = 32 | 32

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 =>

rw [← Nat.cast_ofNat]

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 =>

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

case refine'_2 x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 512) = 32 ⊢ 2 ^ 9 * (2 ^ 10) ^ ↑32 ≤ 4 ^ (512 / 3)

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat]

rw [rpow_nat_cast, ← pow_mul, ← pow_add]

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat]

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

case refine'_2 x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 512) = 32 ⊢ 2 ^ (9 + 10 * 32) ≤ 4 ^ (512 / 3)

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add]

conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add]

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 512) = 32 | 4

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 =>

equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 =>

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 512) = 32 | 4

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 =>

equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 =>

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 512) = 32 | 4

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 =>

equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 =>

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 512) = 32 ⊢ 4 = 2 ^ 2

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) =>

rw [rpow_two]

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) =>

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 512) = 32 ⊢ 4 = 2 ^ 2

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two];

norm_num1

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two];

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

case refine'_2 x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 512) = 32 ⊢ 2 ^ (9 + 10 * 32) ≤ (2 ^ 2) ^ (512 / 3)

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1

rw [← rpow_mul, ← rpow_nat_cast]

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

case refine'_2 x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 512) = 32 ⊢ 2 ^ ↑(9 + 10 * 32) ≤ 2 ^ (2 * (512 / 3)) case refine'_2.hx x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 512) = 32 ⊢ 0 ≤ 2

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast]

apply rpow_le_rpow_of_exponent_le

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast]

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

case refine'_2.hx x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 512) = 32 ⊢ 1 ≤ 2 case refine'_2.hyz x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 512) = 32 ⊢ ↑(9 + 10 * 32) ≤ 2 * (512 / 3) case refine'_2.hx x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 512) = 32 ⊢ 0 ≤ 2

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le

all_goals norm_num1

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

case refine'_2.hx x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 512) = 32 ⊢ 1 ≤ 2

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals

norm_num1

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

case refine'_2.hyz x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 512) = 32 ⊢ ↑(9 + 10 * 32) ≤ 2 * (512 / 3)

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals

norm_num1

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

case refine'_2.hx x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 512) = 32 ⊢ 0 ≤ 2

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals

norm_num1

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

case refine'_2 x : ℝ n_large : 512 ≤ x f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) h5 : 0 < x h : ConcaveOn ℝ (Set.Ioi 0.5) f this : sqrt (2 * 512) = 32 ⊢ 0 < 4 ^ (512 / 3)

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals norm_num1 ·

apply rpow_pos_of_pos four_pos

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals norm_num1 ·

Mathlib.NumberTheory.Bertrand.51_0.gJXoOT9Ce2wC0xc

/-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Mathlib_NumberTheory_Bertrand

n : ℕ n_large : 512 ≤ n ⊢ n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals norm_num1 · apply rpow_pos_of_pos four_pos #align bertrand.real_main_inequality Bertrand.real_main_inequality end Bertrand end Real section Nat open Nat /-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by

rw [← @cast_le ℝ]

/-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by

Mathlib.NumberTheory.Bertrand.121_0.gJXoOT9Ce2wC0xc

/-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n

Mathlib_NumberTheory_Bertrand

n : ℕ n_large : 512 ≤ n ⊢ ↑(n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3)) ≤ ↑(4 ^ n)

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals norm_num1 · apply rpow_pos_of_pos four_pos #align bertrand.real_main_inequality Bertrand.real_main_inequality end Bertrand end Real section Nat open Nat /-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ]

simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast]

/-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ]

Mathlib.NumberTheory.Bertrand.121_0.gJXoOT9Ce2wC0xc

/-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n

Mathlib_NumberTheory_Bertrand

n : ℕ n_large : 512 ≤ n ⊢ ↑n * (↑2 * ↑n) ^ ↑(sqrt (2 * n)) * ↑4 ^ ↑(2 * n / 3) ≤ ↑4 ^ ↑n

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals norm_num1 · apply rpow_pos_of_pos four_pos #align bertrand.real_main_inequality Bertrand.real_main_inequality end Bertrand end Real section Nat open Nat /-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast]

refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large))

/-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast]

Mathlib.NumberTheory.Bertrand.121_0.gJXoOT9Ce2wC0xc

/-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n

Mathlib_NumberTheory_Bertrand

n : ℕ n_large : 512 ≤ n ⊢ 512 ≤ ↑n

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals norm_num1 · apply rpow_pos_of_pos four_pos #align bertrand.real_main_inequality Bertrand.real_main_inequality end Bertrand end Real section Nat open Nat /-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast] refine' _root_.trans ?_ (Bertrand.real_main_inequality (by

exact_mod_cast n_large

/-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast] refine' _root_.trans ?_ (Bertrand.real_main_inequality (by

Mathlib.NumberTheory.Bertrand.121_0.gJXoOT9Ce2wC0xc

/-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n

Mathlib_NumberTheory_Bertrand

n : ℕ n_large : 512 ≤ n ⊢ ↑n * (↑2 * ↑n) ^ ↑(sqrt (2 * n)) * ↑4 ^ ↑(2 * n / 3) ≤ ↑n * (2 * ↑n) ^ Real.sqrt (2 * ↑n) * 4 ^ (2 * ↑n / 3)

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals norm_num1 · apply rpow_pos_of_pos four_pos #align bertrand.real_main_inequality Bertrand.real_main_inequality end Bertrand end Real section Nat open Nat /-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast] refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large))

gcongr

/-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast] refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large))

Mathlib.NumberTheory.Bertrand.121_0.gJXoOT9Ce2wC0xc

/-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n

Mathlib_NumberTheory_Bertrand

case h₁.h.hx n : ℕ n_large : 512 ≤ n ⊢ 1 ≤ ↑2 * ↑n

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals norm_num1 · apply rpow_pos_of_pos four_pos #align bertrand.real_main_inequality Bertrand.real_main_inequality end Bertrand end Real section Nat open Nat /-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast] refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large)) gcongr ·

have n2_pos : 0 < 2 * n := by positivity

/-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast] refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large)) gcongr ·

Mathlib.NumberTheory.Bertrand.121_0.gJXoOT9Ce2wC0xc

/-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n

Mathlib_NumberTheory_Bertrand

n : ℕ n_large : 512 ≤ n ⊢ 0 < 2 * n

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals norm_num1 · apply rpow_pos_of_pos four_pos #align bertrand.real_main_inequality Bertrand.real_main_inequality end Bertrand end Real section Nat open Nat /-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast] refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large)) gcongr · have n2_pos : 0 < 2 * n := by

positivity

/-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast] refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large)) gcongr · have n2_pos : 0 < 2 * n := by

Mathlib.NumberTheory.Bertrand.121_0.gJXoOT9Ce2wC0xc

/-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n

Mathlib_NumberTheory_Bertrand

case h₁.h.hx n : ℕ n_large : 512 ≤ n n2_pos : 0 < 2 * n ⊢ 1 ≤ ↑2 * ↑n

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals norm_num1 · apply rpow_pos_of_pos four_pos #align bertrand.real_main_inequality Bertrand.real_main_inequality end Bertrand end Real section Nat open Nat /-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast] refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large)) gcongr · have n2_pos : 0 < 2 * n := by positivity

exact mod_cast n2_pos

/-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast] refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large)) gcongr · have n2_pos : 0 < 2 * n := by positivity

Mathlib.NumberTheory.Bertrand.121_0.gJXoOT9Ce2wC0xc

/-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n

Mathlib_NumberTheory_Bertrand

case h₁.h.hyz n : ℕ n_large : 512 ≤ n ⊢ ↑(sqrt (2 * n)) ≤ Real.sqrt (2 * ↑n)

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals norm_num1 · apply rpow_pos_of_pos four_pos #align bertrand.real_main_inequality Bertrand.real_main_inequality end Bertrand end Real section Nat open Nat /-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast] refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large)) gcongr · have n2_pos : 0 < 2 * n := by positivity exact mod_cast n2_pos ·

exact_mod_cast Real.nat_sqrt_le_real_sqrt

/-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast] refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large)) gcongr · have n2_pos : 0 < 2 * n := by positivity exact mod_cast n2_pos ·

Mathlib.NumberTheory.Bertrand.121_0.gJXoOT9Ce2wC0xc

/-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n

Mathlib_NumberTheory_Bertrand

case h₂.hx n : ℕ n_large : 512 ≤ n ⊢ 1 ≤ ↑4

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals norm_num1 · apply rpow_pos_of_pos four_pos #align bertrand.real_main_inequality Bertrand.real_main_inequality end Bertrand end Real section Nat open Nat /-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast] refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large)) gcongr · have n2_pos : 0 < 2 * n := by positivity exact mod_cast n2_pos · exact_mod_cast Real.nat_sqrt_le_real_sqrt ·

norm_num1

/-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast] refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large)) gcongr · have n2_pos : 0 < 2 * n := by positivity exact mod_cast n2_pos · exact_mod_cast Real.nat_sqrt_le_real_sqrt ·

Mathlib.NumberTheory.Bertrand.121_0.gJXoOT9Ce2wC0xc

/-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n

Mathlib_NumberTheory_Bertrand

case h₂.hyz n : ℕ n_large : 512 ≤ n ⊢ ↑(2 * n / 3) ≤ 2 * ↑n / 3

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals norm_num1 · apply rpow_pos_of_pos four_pos #align bertrand.real_main_inequality Bertrand.real_main_inequality end Bertrand end Real section Nat open Nat /-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast] refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large)) gcongr · have n2_pos : 0 < 2 * n := by positivity exact mod_cast n2_pos · exact_mod_cast Real.nat_sqrt_le_real_sqrt · norm_num1 ·

exact cast_div_le.trans (by norm_cast)

/-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast] refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large)) gcongr · have n2_pos : 0 < 2 * n := by positivity exact mod_cast n2_pos · exact_mod_cast Real.nat_sqrt_le_real_sqrt · norm_num1 ·

Mathlib.NumberTheory.Bertrand.121_0.gJXoOT9Ce2wC0xc

/-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n

Mathlib_NumberTheory_Bertrand

n : ℕ n_large : 512 ≤ n ⊢ ↑(2 * n) / ↑3 ≤ 2 * ↑n / 3

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals norm_num1 · apply rpow_pos_of_pos four_pos #align bertrand.real_main_inequality Bertrand.real_main_inequality end Bertrand end Real section Nat open Nat /-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast] refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large)) gcongr · have n2_pos : 0 < 2 * n := by positivity exact mod_cast n2_pos · exact_mod_cast Real.nat_sqrt_le_real_sqrt · norm_num1 · exact cast_div_le.trans (by

norm_cast

/-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast] refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large)) gcongr · have n2_pos : 0 < 2 * n := by positivity exact mod_cast n2_pos · exact_mod_cast Real.nat_sqrt_le_real_sqrt · norm_num1 · exact cast_div_le.trans (by

Mathlib.NumberTheory.Bertrand.121_0.gJXoOT9Ce2wC0xc

/-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n

Mathlib_NumberTheory_Bertrand

n : ℕ n_large : 2 < n no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n ⊢ centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (Nat.factorization (centralBinom n)) p

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals norm_num1 · apply rpow_pos_of_pos four_pos #align bertrand.real_main_inequality Bertrand.real_main_inequality end Bertrand end Real section Nat open Nat /-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast] refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large)) gcongr · have n2_pos : 0 < 2 * n := by positivity exact mod_cast n2_pos · exact_mod_cast Real.nat_sqrt_le_real_sqrt · norm_num1 · exact cast_div_le.trans (by norm_cast) #align bertrand_main_inequality bertrand_main_inequality /-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by

refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm

/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by

Mathlib.NumberTheory.Bertrand.136_0.gJXoOT9Ce2wC0xc

/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p

Mathlib_NumberTheory_Bertrand

n : ℕ n_large : 2 < n no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n ⊢ ∏ p in Finset.range (2 * n / 3 + 1), p ^ (Nat.factorization (centralBinom n)) p = ∏ p in Finset.range (2 * n + 1), p ^ (Nat.factorization (centralBinom n)) p

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals norm_num1 · apply rpow_pos_of_pos four_pos #align bertrand.real_main_inequality Bertrand.real_main_inequality end Bertrand end Real section Nat open Nat /-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast] refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large)) gcongr · have n2_pos : 0 < 2 * n := by positivity exact mod_cast n2_pos · exact_mod_cast Real.nat_sqrt_le_real_sqrt · norm_num1 · exact cast_div_le.trans (by norm_cast) #align bertrand_main_inequality bertrand_main_inequality /-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm

apply Finset.prod_subset

/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm

Mathlib.NumberTheory.Bertrand.136_0.gJXoOT9Ce2wC0xc

/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p

Mathlib_NumberTheory_Bertrand

case h n : ℕ n_large : 2 < n no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n ⊢ Finset.range (2 * n / 3 + 1) ⊆ Finset.range (2 * n + 1)

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals norm_num1 · apply rpow_pos_of_pos four_pos #align bertrand.real_main_inequality Bertrand.real_main_inequality end Bertrand end Real section Nat open Nat /-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast] refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large)) gcongr · have n2_pos : 0 < 2 * n := by positivity exact mod_cast n2_pos · exact_mod_cast Real.nat_sqrt_le_real_sqrt · norm_num1 · exact cast_div_le.trans (by norm_cast) #align bertrand_main_inequality bertrand_main_inequality /-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm apply Finset.prod_subset ·

exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _)

/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm apply Finset.prod_subset ·

Mathlib.NumberTheory.Bertrand.136_0.gJXoOT9Ce2wC0xc

/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p

Mathlib_NumberTheory_Bertrand

case hf n : ℕ n_large : 2 < n no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n ⊢ ∀ x ∈ Finset.range (2 * n + 1), x ∉ Finset.range (2 * n / 3 + 1) → x ^ (Nat.factorization (centralBinom n)) x = 1

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals norm_num1 · apply rpow_pos_of_pos four_pos #align bertrand.real_main_inequality Bertrand.real_main_inequality end Bertrand end Real section Nat open Nat /-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast] refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large)) gcongr · have n2_pos : 0 < 2 * n := by positivity exact mod_cast n2_pos · exact_mod_cast Real.nat_sqrt_le_real_sqrt · norm_num1 · exact cast_div_le.trans (by norm_cast) #align bertrand_main_inequality bertrand_main_inequality /-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm apply Finset.prod_subset · exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _)

intro x hx h2x

/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm apply Finset.prod_subset · exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _)

Mathlib.NumberTheory.Bertrand.136_0.gJXoOT9Ce2wC0xc

/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p

Mathlib_NumberTheory_Bertrand

case hf n : ℕ n_large : 2 < n no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n x : ℕ hx : x ∈ Finset.range (2 * n + 1) h2x : x ∉ Finset.range (2 * n / 3 + 1) ⊢ x ^ (Nat.factorization (centralBinom n)) x = 1

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals norm_num1 · apply rpow_pos_of_pos four_pos #align bertrand.real_main_inequality Bertrand.real_main_inequality end Bertrand end Real section Nat open Nat /-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast] refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large)) gcongr · have n2_pos : 0 < 2 * n := by positivity exact mod_cast n2_pos · exact_mod_cast Real.nat_sqrt_le_real_sqrt · norm_num1 · exact cast_div_le.trans (by norm_cast) #align bertrand_main_inequality bertrand_main_inequality /-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm apply Finset.prod_subset · exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _) intro x hx h2x

rw [Finset.mem_range, lt_succ_iff] at hx h2x

/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm apply Finset.prod_subset · exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _) intro x hx h2x

Mathlib.NumberTheory.Bertrand.136_0.gJXoOT9Ce2wC0xc

/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p

Mathlib_NumberTheory_Bertrand

case hf n : ℕ n_large : 2 < n no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n x : ℕ hx : x ≤ 2 * n h2x : ¬x ≤ 2 * n / 3 ⊢ x ^ (Nat.factorization (centralBinom n)) x = 1

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals norm_num1 · apply rpow_pos_of_pos four_pos #align bertrand.real_main_inequality Bertrand.real_main_inequality end Bertrand end Real section Nat open Nat /-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast] refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large)) gcongr · have n2_pos : 0 < 2 * n := by positivity exact mod_cast n2_pos · exact_mod_cast Real.nat_sqrt_le_real_sqrt · norm_num1 · exact cast_div_le.trans (by norm_cast) #align bertrand_main_inequality bertrand_main_inequality /-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm apply Finset.prod_subset · exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _) intro x hx h2x rw [Finset.mem_range, lt_succ_iff] at hx h2x

rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x

/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm apply Finset.prod_subset · exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _) intro x hx h2x rw [Finset.mem_range, lt_succ_iff] at hx h2x

Mathlib.NumberTheory.Bertrand.136_0.gJXoOT9Ce2wC0xc

/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p

Mathlib_NumberTheory_Bertrand

case hf n : ℕ n_large : 2 < n no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n x : ℕ hx : x ≤ 2 * n h2x : 2 * n < 3 * x ⊢ x ^ (Nat.factorization (centralBinom n)) x = 1

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals norm_num1 · apply rpow_pos_of_pos four_pos #align bertrand.real_main_inequality Bertrand.real_main_inequality end Bertrand end Real section Nat open Nat /-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast] refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large)) gcongr · have n2_pos : 0 < 2 * n := by positivity exact mod_cast n2_pos · exact_mod_cast Real.nat_sqrt_le_real_sqrt · norm_num1 · exact cast_div_le.trans (by norm_cast) #align bertrand_main_inequality bertrand_main_inequality /-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm apply Finset.prod_subset · exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _) intro x hx h2x rw [Finset.mem_range, lt_succ_iff] at hx h2x rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x

replace no_prime := not_exists.mp no_prime x

/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm apply Finset.prod_subset · exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _) intro x hx h2x rw [Finset.mem_range, lt_succ_iff] at hx h2x rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x

Mathlib.NumberTheory.Bertrand.136_0.gJXoOT9Ce2wC0xc

/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p

Mathlib_NumberTheory_Bertrand

case hf n : ℕ n_large : 2 < n x : ℕ hx : x ≤ 2 * n h2x : 2 * n < 3 * x no_prime : ¬(Nat.Prime x ∧ n < x ∧ x ≤ 2 * n) ⊢ x ^ (Nat.factorization (centralBinom n)) x = 1

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals norm_num1 · apply rpow_pos_of_pos four_pos #align bertrand.real_main_inequality Bertrand.real_main_inequality end Bertrand end Real section Nat open Nat /-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast] refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large)) gcongr · have n2_pos : 0 < 2 * n := by positivity exact mod_cast n2_pos · exact_mod_cast Real.nat_sqrt_le_real_sqrt · norm_num1 · exact cast_div_le.trans (by norm_cast) #align bertrand_main_inequality bertrand_main_inequality /-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm apply Finset.prod_subset · exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _) intro x hx h2x rw [Finset.mem_range, lt_succ_iff] at hx h2x rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x replace no_prime := not_exists.mp no_prime x

rw [← and_assoc, not_and', not_and_or, not_lt] at no_prime

/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm apply Finset.prod_subset · exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _) intro x hx h2x rw [Finset.mem_range, lt_succ_iff] at hx h2x rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x replace no_prime := not_exists.mp no_prime x

Mathlib.NumberTheory.Bertrand.136_0.gJXoOT9Ce2wC0xc

/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p

Mathlib_NumberTheory_Bertrand

case hf n : ℕ n_large : 2 < n x : ℕ hx : x ≤ 2 * n h2x : 2 * n < 3 * x no_prime : x ≤ 2 * n → ¬Nat.Prime x ∨ x ≤ n ⊢ x ^ (Nat.factorization (centralBinom n)) x = 1

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals norm_num1 · apply rpow_pos_of_pos four_pos #align bertrand.real_main_inequality Bertrand.real_main_inequality end Bertrand end Real section Nat open Nat /-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast] refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large)) gcongr · have n2_pos : 0 < 2 * n := by positivity exact mod_cast n2_pos · exact_mod_cast Real.nat_sqrt_le_real_sqrt · norm_num1 · exact cast_div_le.trans (by norm_cast) #align bertrand_main_inequality bertrand_main_inequality /-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm apply Finset.prod_subset · exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _) intro x hx h2x rw [Finset.mem_range, lt_succ_iff] at hx h2x rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x replace no_prime := not_exists.mp no_prime x rw [← and_assoc, not_and', not_and_or, not_lt] at no_prime

cases' no_prime hx with h h

/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm apply Finset.prod_subset · exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _) intro x hx h2x rw [Finset.mem_range, lt_succ_iff] at hx h2x rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x replace no_prime := not_exists.mp no_prime x rw [← and_assoc, not_and', not_and_or, not_lt] at no_prime

Mathlib.NumberTheory.Bertrand.136_0.gJXoOT9Ce2wC0xc

/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p

Mathlib_NumberTheory_Bertrand

case hf.inl n : ℕ n_large : 2 < n x : ℕ hx : x ≤ 2 * n h2x : 2 * n < 3 * x no_prime : x ≤ 2 * n → ¬Nat.Prime x ∨ x ≤ n h : ¬Nat.Prime x ⊢ x ^ (Nat.factorization (centralBinom n)) x = 1

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals norm_num1 · apply rpow_pos_of_pos four_pos #align bertrand.real_main_inequality Bertrand.real_main_inequality end Bertrand end Real section Nat open Nat /-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast] refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large)) gcongr · have n2_pos : 0 < 2 * n := by positivity exact mod_cast n2_pos · exact_mod_cast Real.nat_sqrt_le_real_sqrt · norm_num1 · exact cast_div_le.trans (by norm_cast) #align bertrand_main_inequality bertrand_main_inequality /-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm apply Finset.prod_subset · exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _) intro x hx h2x rw [Finset.mem_range, lt_succ_iff] at hx h2x rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x replace no_prime := not_exists.mp no_prime x rw [← and_assoc, not_and', not_and_or, not_lt] at no_prime cases' no_prime hx with h h ·

rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero]

/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm apply Finset.prod_subset · exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _) intro x hx h2x rw [Finset.mem_range, lt_succ_iff] at hx h2x rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x replace no_prime := not_exists.mp no_prime x rw [← and_assoc, not_and', not_and_or, not_lt] at no_prime cases' no_prime hx with h h ·

Mathlib.NumberTheory.Bertrand.136_0.gJXoOT9Ce2wC0xc

/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p

Mathlib_NumberTheory_Bertrand

case hf.inr n : ℕ n_large : 2 < n x : ℕ hx : x ≤ 2 * n h2x : 2 * n < 3 * x no_prime : x ≤ 2 * n → ¬Nat.Prime x ∨ x ≤ n h : x ≤ n ⊢ x ^ (Nat.factorization (centralBinom n)) x = 1

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals norm_num1 · apply rpow_pos_of_pos four_pos #align bertrand.real_main_inequality Bertrand.real_main_inequality end Bertrand end Real section Nat open Nat /-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast] refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large)) gcongr · have n2_pos : 0 < 2 * n := by positivity exact mod_cast n2_pos · exact_mod_cast Real.nat_sqrt_le_real_sqrt · norm_num1 · exact cast_div_le.trans (by norm_cast) #align bertrand_main_inequality bertrand_main_inequality /-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm apply Finset.prod_subset · exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _) intro x hx h2x rw [Finset.mem_range, lt_succ_iff] at hx h2x rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x replace no_prime := not_exists.mp no_prime x rw [← and_assoc, not_and', not_and_or, not_lt] at no_prime cases' no_prime hx with h h · rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero] ·

rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero]

/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm apply Finset.prod_subset · exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _) intro x hx h2x rw [Finset.mem_range, lt_succ_iff] at hx h2x rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x replace no_prime := not_exists.mp no_prime x rw [← and_assoc, not_and', not_and_or, not_lt] at no_prime cases' no_prime hx with h h · rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero] ·

Mathlib.NumberTheory.Bertrand.136_0.gJXoOT9Ce2wC0xc

/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p

Mathlib_NumberTheory_Bertrand

n : ℕ n_big : 2 < n no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n ⊢ centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3)

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals norm_num1 · apply rpow_pos_of_pos four_pos #align bertrand.real_main_inequality Bertrand.real_main_inequality end Bertrand end Real section Nat open Nat /-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast] refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large)) gcongr · have n2_pos : 0 < 2 * n := by positivity exact mod_cast n2_pos · exact_mod_cast Real.nat_sqrt_le_real_sqrt · norm_num1 · exact cast_div_le.trans (by norm_cast) #align bertrand_main_inequality bertrand_main_inequality /-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm apply Finset.prod_subset · exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _) intro x hx h2x rw [Finset.mem_range, lt_succ_iff] at hx h2x rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x replace no_prime := not_exists.mp no_prime x rw [← and_assoc, not_and', not_and_or, not_lt] at no_prime cases' no_prime hx with h h · rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero] · rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero] #align central_binom_factorization_small centralBinom_factorization_small /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate. The bound splits the prime factors of `centralBinom n` into those 1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime. 2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total. 3. Between `2 * n / 3` and `n`, which do not exist. 4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false. 5. Above `2 * n`, which do not exist. -/ theorem centralBinom_le_of_no_bertrand_prime (n : ℕ) (n_big : 2 < n) (no_prime : ¬∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n) : centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by

have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big

/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate. The bound splits the prime factors of `centralBinom n` into those 1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime. 2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total. 3. Between `2 * n / 3` and `n`, which do not exist. 4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false. 5. Above `2 * n`, which do not exist. -/ theorem centralBinom_le_of_no_bertrand_prime (n : ℕ) (n_big : 2 < n) (no_prime : ¬∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n) : centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by

Mathlib.NumberTheory.Bertrand.155_0.gJXoOT9Ce2wC0xc

/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate. The bound splits the prime factors of `centralBinom n` into those 1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime. 2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total. 3. Between `2 * n / 3` and `n`, which do not exist. 4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false. 5. Above `2 * n`, which do not exist. -/ theorem centralBinom_le_of_no_bertrand_prime (n : ℕ) (n_big : 2 < n) (no_prime : ¬∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n) : centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3)

Mathlib_NumberTheory_Bertrand

n : ℕ n_big : 2 < n no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n n_pos : 0 < n ⊢ centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3)

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals norm_num1 · apply rpow_pos_of_pos four_pos #align bertrand.real_main_inequality Bertrand.real_main_inequality end Bertrand end Real section Nat open Nat /-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast] refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large)) gcongr · have n2_pos : 0 < 2 * n := by positivity exact mod_cast n2_pos · exact_mod_cast Real.nat_sqrt_le_real_sqrt · norm_num1 · exact cast_div_le.trans (by norm_cast) #align bertrand_main_inequality bertrand_main_inequality /-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm apply Finset.prod_subset · exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _) intro x hx h2x rw [Finset.mem_range, lt_succ_iff] at hx h2x rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x replace no_prime := not_exists.mp no_prime x rw [← and_assoc, not_and', not_and_or, not_lt] at no_prime cases' no_prime hx with h h · rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero] · rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero] #align central_binom_factorization_small centralBinom_factorization_small /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate. The bound splits the prime factors of `centralBinom n` into those 1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime. 2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total. 3. Between `2 * n / 3` and `n`, which do not exist. 4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false. 5. Above `2 * n`, which do not exist. -/ theorem centralBinom_le_of_no_bertrand_prime (n : ℕ) (n_big : 2 < n) (no_prime : ¬∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n) : centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big

have n2_pos : 1 ≤ 2 * n := mul_pos (zero_lt_two' ℕ) n_pos

/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate. The bound splits the prime factors of `centralBinom n` into those 1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime. 2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total. 3. Between `2 * n / 3` and `n`, which do not exist. 4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false. 5. Above `2 * n`, which do not exist. -/ theorem centralBinom_le_of_no_bertrand_prime (n : ℕ) (n_big : 2 < n) (no_prime : ¬∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n) : centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big

Mathlib.NumberTheory.Bertrand.155_0.gJXoOT9Ce2wC0xc

/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate. The bound splits the prime factors of `centralBinom n` into those 1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime. 2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total. 3. Between `2 * n / 3` and `n`, which do not exist. 4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false. 5. Above `2 * n`, which do not exist. -/ theorem centralBinom_le_of_no_bertrand_prime (n : ℕ) (n_big : 2 < n) (no_prime : ¬∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n) : centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3)

Mathlib_NumberTheory_Bertrand

n : ℕ n_big : 2 < n no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n n_pos : 0 < n n2_pos : 1 ≤ 2 * n ⊢ centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3)

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals norm_num1 · apply rpow_pos_of_pos four_pos #align bertrand.real_main_inequality Bertrand.real_main_inequality end Bertrand end Real section Nat open Nat /-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast] refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large)) gcongr · have n2_pos : 0 < 2 * n := by positivity exact mod_cast n2_pos · exact_mod_cast Real.nat_sqrt_le_real_sqrt · norm_num1 · exact cast_div_le.trans (by norm_cast) #align bertrand_main_inequality bertrand_main_inequality /-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm apply Finset.prod_subset · exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _) intro x hx h2x rw [Finset.mem_range, lt_succ_iff] at hx h2x rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x replace no_prime := not_exists.mp no_prime x rw [← and_assoc, not_and', not_and_or, not_lt] at no_prime cases' no_prime hx with h h · rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero] · rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero] #align central_binom_factorization_small centralBinom_factorization_small /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate. The bound splits the prime factors of `centralBinom n` into those 1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime. 2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total. 3. Between `2 * n / 3` and `n`, which do not exist. 4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false. 5. Above `2 * n`, which do not exist. -/ theorem centralBinom_le_of_no_bertrand_prime (n : ℕ) (n_big : 2 < n) (no_prime : ¬∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n) : centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big have n2_pos : 1 ≤ 2 * n := mul_pos (zero_lt_two' ℕ) n_pos

let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime

/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate. The bound splits the prime factors of `centralBinom n` into those 1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime. 2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total. 3. Between `2 * n / 3` and `n`, which do not exist. 4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false. 5. Above `2 * n`, which do not exist. -/ theorem centralBinom_le_of_no_bertrand_prime (n : ℕ) (n_big : 2 < n) (no_prime : ¬∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n) : centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big have n2_pos : 1 ≤ 2 * n := mul_pos (zero_lt_two' ℕ) n_pos

Mathlib.NumberTheory.Bertrand.155_0.gJXoOT9Ce2wC0xc

/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate. The bound splits the prime factors of `centralBinom n` into those 1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime. 2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total. 3. Between `2 * n / 3` and `n`, which do not exist. 4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false. 5. Above `2 * n`, which do not exist. -/ theorem centralBinom_le_of_no_bertrand_prime (n : ℕ) (n_big : 2 < n) (no_prime : ¬∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n) : centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3)

Mathlib_NumberTheory_Bertrand

n : ℕ n_big : 2 < n no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n n_pos : 0 < n n2_pos : 1 ≤ 2 * n S : Finset ℕ := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1)) ⊢ centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3)

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals norm_num1 · apply rpow_pos_of_pos four_pos #align bertrand.real_main_inequality Bertrand.real_main_inequality end Bertrand end Real section Nat open Nat /-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast] refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large)) gcongr · have n2_pos : 0 < 2 * n := by positivity exact mod_cast n2_pos · exact_mod_cast Real.nat_sqrt_le_real_sqrt · norm_num1 · exact cast_div_le.trans (by norm_cast) #align bertrand_main_inequality bertrand_main_inequality /-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm apply Finset.prod_subset · exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _) intro x hx h2x rw [Finset.mem_range, lt_succ_iff] at hx h2x rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x replace no_prime := not_exists.mp no_prime x rw [← and_assoc, not_and', not_and_or, not_lt] at no_prime cases' no_prime hx with h h · rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero] · rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero] #align central_binom_factorization_small centralBinom_factorization_small /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate. The bound splits the prime factors of `centralBinom n` into those 1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime. 2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total. 3. Between `2 * n / 3` and `n`, which do not exist. 4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false. 5. Above `2 * n`, which do not exist. -/ theorem centralBinom_le_of_no_bertrand_prime (n : ℕ) (n_big : 2 < n) (no_prime : ¬∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n) : centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big have n2_pos : 1 ≤ 2 * n := mul_pos (zero_lt_two' ℕ) n_pos let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime

let f x := x ^ n.centralBinom.factorization x

/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate. The bound splits the prime factors of `centralBinom n` into those 1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime. 2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total. 3. Between `2 * n / 3` and `n`, which do not exist. 4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false. 5. Above `2 * n`, which do not exist. -/ theorem centralBinom_le_of_no_bertrand_prime (n : ℕ) (n_big : 2 < n) (no_prime : ¬∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n) : centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big have n2_pos : 1 ≤ 2 * n := mul_pos (zero_lt_two' ℕ) n_pos let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime

Mathlib.NumberTheory.Bertrand.155_0.gJXoOT9Ce2wC0xc

/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate. The bound splits the prime factors of `centralBinom n` into those 1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime. 2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total. 3. Between `2 * n / 3` and `n`, which do not exist. 4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false. 5. Above `2 * n`, which do not exist. -/ theorem centralBinom_le_of_no_bertrand_prime (n : ℕ) (n_big : 2 < n) (no_prime : ¬∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n) : centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3)

Mathlib_NumberTheory_Bertrand

n : ℕ n_big : 2 < n no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n n_pos : 0 < n n2_pos : 1 ≤ 2 * n S : Finset ℕ := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1)) f : ℕ → ℕ := fun x => x ^ (Nat.factorization (centralBinom n)) x ⊢ centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3)

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals norm_num1 · apply rpow_pos_of_pos four_pos #align bertrand.real_main_inequality Bertrand.real_main_inequality end Bertrand end Real section Nat open Nat /-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast] refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large)) gcongr · have n2_pos : 0 < 2 * n := by positivity exact mod_cast n2_pos · exact_mod_cast Real.nat_sqrt_le_real_sqrt · norm_num1 · exact cast_div_le.trans (by norm_cast) #align bertrand_main_inequality bertrand_main_inequality /-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm apply Finset.prod_subset · exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _) intro x hx h2x rw [Finset.mem_range, lt_succ_iff] at hx h2x rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x replace no_prime := not_exists.mp no_prime x rw [← and_assoc, not_and', not_and_or, not_lt] at no_prime cases' no_prime hx with h h · rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero] · rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero] #align central_binom_factorization_small centralBinom_factorization_small /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate. The bound splits the prime factors of `centralBinom n` into those 1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime. 2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total. 3. Between `2 * n / 3` and `n`, which do not exist. 4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false. 5. Above `2 * n`, which do not exist. -/ theorem centralBinom_le_of_no_bertrand_prime (n : ℕ) (n_big : 2 < n) (no_prime : ¬∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n) : centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big have n2_pos : 1 ≤ 2 * n := mul_pos (zero_lt_two' ℕ) n_pos let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime let f x := x ^ n.centralBinom.factorization x

have : ∏ x : ℕ in S, f x = ∏ x : ℕ in Finset.range (2 * n / 3 + 1), f x := by refine' Finset.prod_filter_of_ne fun p _ h => _ contrapose! h; dsimp only rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]

/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate. The bound splits the prime factors of `centralBinom n` into those 1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime. 2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total. 3. Between `2 * n / 3` and `n`, which do not exist. 4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false. 5. Above `2 * n`, which do not exist. -/ theorem centralBinom_le_of_no_bertrand_prime (n : ℕ) (n_big : 2 < n) (no_prime : ¬∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n) : centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big have n2_pos : 1 ≤ 2 * n := mul_pos (zero_lt_two' ℕ) n_pos let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime let f x := x ^ n.centralBinom.factorization x

Mathlib.NumberTheory.Bertrand.155_0.gJXoOT9Ce2wC0xc

/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate. The bound splits the prime factors of `centralBinom n` into those 1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime. 2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total. 3. Between `2 * n / 3` and `n`, which do not exist. 4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false. 5. Above `2 * n`, which do not exist. -/ theorem centralBinom_le_of_no_bertrand_prime (n : ℕ) (n_big : 2 < n) (no_prime : ¬∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n) : centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3)

Mathlib_NumberTheory_Bertrand

n : ℕ n_big : 2 < n no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n n_pos : 0 < n n2_pos : 1 ≤ 2 * n S : Finset ℕ := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1)) f : ℕ → ℕ := fun x => x ^ (Nat.factorization (centralBinom n)) x ⊢ ∏ x in S, f x = ∏ x in Finset.range (2 * n / 3 + 1), f x

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals norm_num1 · apply rpow_pos_of_pos four_pos #align bertrand.real_main_inequality Bertrand.real_main_inequality end Bertrand end Real section Nat open Nat /-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast] refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large)) gcongr · have n2_pos : 0 < 2 * n := by positivity exact mod_cast n2_pos · exact_mod_cast Real.nat_sqrt_le_real_sqrt · norm_num1 · exact cast_div_le.trans (by norm_cast) #align bertrand_main_inequality bertrand_main_inequality /-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm apply Finset.prod_subset · exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _) intro x hx h2x rw [Finset.mem_range, lt_succ_iff] at hx h2x rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x replace no_prime := not_exists.mp no_prime x rw [← and_assoc, not_and', not_and_or, not_lt] at no_prime cases' no_prime hx with h h · rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero] · rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero] #align central_binom_factorization_small centralBinom_factorization_small /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate. The bound splits the prime factors of `centralBinom n` into those 1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime. 2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total. 3. Between `2 * n / 3` and `n`, which do not exist. 4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false. 5. Above `2 * n`, which do not exist. -/ theorem centralBinom_le_of_no_bertrand_prime (n : ℕ) (n_big : 2 < n) (no_prime : ¬∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n) : centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big have n2_pos : 1 ≤ 2 * n := mul_pos (zero_lt_two' ℕ) n_pos let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime let f x := x ^ n.centralBinom.factorization x have : ∏ x : ℕ in S, f x = ∏ x : ℕ in Finset.range (2 * n / 3 + 1), f x := by

refine' Finset.prod_filter_of_ne fun p _ h => _

/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate. The bound splits the prime factors of `centralBinom n` into those 1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime. 2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total. 3. Between `2 * n / 3` and `n`, which do not exist. 4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false. 5. Above `2 * n`, which do not exist. -/ theorem centralBinom_le_of_no_bertrand_prime (n : ℕ) (n_big : 2 < n) (no_prime : ¬∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n) : centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big have n2_pos : 1 ≤ 2 * n := mul_pos (zero_lt_two' ℕ) n_pos let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime let f x := x ^ n.centralBinom.factorization x have : ∏ x : ℕ in S, f x = ∏ x : ℕ in Finset.range (2 * n / 3 + 1), f x := by

Mathlib.NumberTheory.Bertrand.155_0.gJXoOT9Ce2wC0xc

/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate. The bound splits the prime factors of `centralBinom n` into those 1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime. 2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total. 3. Between `2 * n / 3` and `n`, which do not exist. 4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false. 5. Above `2 * n`, which do not exist. -/ theorem centralBinom_le_of_no_bertrand_prime (n : ℕ) (n_big : 2 < n) (no_prime : ¬∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n) : centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3)

Mathlib_NumberTheory_Bertrand

n : ℕ n_big : 2 < n no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n n_pos : 0 < n n2_pos : 1 ≤ 2 * n S : Finset ℕ := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1)) f : ℕ → ℕ := fun x => x ^ (Nat.factorization (centralBinom n)) x p : ℕ x✝ : p ∈ Finset.range (2 * n / 3 + 1) h : f p ≠ 1 ⊢ Nat.Prime p

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals norm_num1 · apply rpow_pos_of_pos four_pos #align bertrand.real_main_inequality Bertrand.real_main_inequality end Bertrand end Real section Nat open Nat /-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast] refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large)) gcongr · have n2_pos : 0 < 2 * n := by positivity exact mod_cast n2_pos · exact_mod_cast Real.nat_sqrt_le_real_sqrt · norm_num1 · exact cast_div_le.trans (by norm_cast) #align bertrand_main_inequality bertrand_main_inequality /-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm apply Finset.prod_subset · exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _) intro x hx h2x rw [Finset.mem_range, lt_succ_iff] at hx h2x rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x replace no_prime := not_exists.mp no_prime x rw [← and_assoc, not_and', not_and_or, not_lt] at no_prime cases' no_prime hx with h h · rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero] · rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero] #align central_binom_factorization_small centralBinom_factorization_small /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate. The bound splits the prime factors of `centralBinom n` into those 1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime. 2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total. 3. Between `2 * n / 3` and `n`, which do not exist. 4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false. 5. Above `2 * n`, which do not exist. -/ theorem centralBinom_le_of_no_bertrand_prime (n : ℕ) (n_big : 2 < n) (no_prime : ¬∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n) : centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big have n2_pos : 1 ≤ 2 * n := mul_pos (zero_lt_two' ℕ) n_pos let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime let f x := x ^ n.centralBinom.factorization x have : ∏ x : ℕ in S, f x = ∏ x : ℕ in Finset.range (2 * n / 3 + 1), f x := by refine' Finset.prod_filter_of_ne fun p _ h => _

contrapose! h

/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate. The bound splits the prime factors of `centralBinom n` into those 1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime. 2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total. 3. Between `2 * n / 3` and `n`, which do not exist. 4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false. 5. Above `2 * n`, which do not exist. -/ theorem centralBinom_le_of_no_bertrand_prime (n : ℕ) (n_big : 2 < n) (no_prime : ¬∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n) : centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big have n2_pos : 1 ≤ 2 * n := mul_pos (zero_lt_two' ℕ) n_pos let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime let f x := x ^ n.centralBinom.factorization x have : ∏ x : ℕ in S, f x = ∏ x : ℕ in Finset.range (2 * n / 3 + 1), f x := by refine' Finset.prod_filter_of_ne fun p _ h => _

Mathlib.NumberTheory.Bertrand.155_0.gJXoOT9Ce2wC0xc

/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate. The bound splits the prime factors of `centralBinom n` into those 1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime. 2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total. 3. Between `2 * n / 3` and `n`, which do not exist. 4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false. 5. Above `2 * n`, which do not exist. -/ theorem centralBinom_le_of_no_bertrand_prime (n : ℕ) (n_big : 2 < n) (no_prime : ¬∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n) : centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3)

Mathlib_NumberTheory_Bertrand

n : ℕ n_big : 2 < n no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n n_pos : 0 < n n2_pos : 1 ≤ 2 * n S : Finset ℕ := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1)) f : ℕ → ℕ := fun x => x ^ (Nat.factorization (centralBinom n)) x p : ℕ x✝ : p ∈ Finset.range (2 * n / 3 + 1) h : ¬Nat.Prime p ⊢ f p = 1

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals norm_num1 · apply rpow_pos_of_pos four_pos #align bertrand.real_main_inequality Bertrand.real_main_inequality end Bertrand end Real section Nat open Nat /-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast] refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large)) gcongr · have n2_pos : 0 < 2 * n := by positivity exact mod_cast n2_pos · exact_mod_cast Real.nat_sqrt_le_real_sqrt · norm_num1 · exact cast_div_le.trans (by norm_cast) #align bertrand_main_inequality bertrand_main_inequality /-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm apply Finset.prod_subset · exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _) intro x hx h2x rw [Finset.mem_range, lt_succ_iff] at hx h2x rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x replace no_prime := not_exists.mp no_prime x rw [← and_assoc, not_and', not_and_or, not_lt] at no_prime cases' no_prime hx with h h · rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero] · rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero] #align central_binom_factorization_small centralBinom_factorization_small /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate. The bound splits the prime factors of `centralBinom n` into those 1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime. 2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total. 3. Between `2 * n / 3` and `n`, which do not exist. 4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false. 5. Above `2 * n`, which do not exist. -/ theorem centralBinom_le_of_no_bertrand_prime (n : ℕ) (n_big : 2 < n) (no_prime : ¬∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n) : centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big have n2_pos : 1 ≤ 2 * n := mul_pos (zero_lt_two' ℕ) n_pos let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime let f x := x ^ n.centralBinom.factorization x have : ∏ x : ℕ in S, f x = ∏ x : ℕ in Finset.range (2 * n / 3 + 1), f x := by refine' Finset.prod_filter_of_ne fun p _ h => _ contrapose! h;

dsimp only

/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate. The bound splits the prime factors of `centralBinom n` into those 1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime. 2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total. 3. Between `2 * n / 3` and `n`, which do not exist. 4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false. 5. Above `2 * n`, which do not exist. -/ theorem centralBinom_le_of_no_bertrand_prime (n : ℕ) (n_big : 2 < n) (no_prime : ¬∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n) : centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big have n2_pos : 1 ≤ 2 * n := mul_pos (zero_lt_two' ℕ) n_pos let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime let f x := x ^ n.centralBinom.factorization x have : ∏ x : ℕ in S, f x = ∏ x : ℕ in Finset.range (2 * n / 3 + 1), f x := by refine' Finset.prod_filter_of_ne fun p _ h => _ contrapose! h;

Mathlib.NumberTheory.Bertrand.155_0.gJXoOT9Ce2wC0xc

/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate. The bound splits the prime factors of `centralBinom n` into those 1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime. 2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total. 3. Between `2 * n / 3` and `n`, which do not exist. 4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false. 5. Above `2 * n`, which do not exist. -/ theorem centralBinom_le_of_no_bertrand_prime (n : ℕ) (n_big : 2 < n) (no_prime : ¬∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n) : centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3)

Mathlib_NumberTheory_Bertrand

n : ℕ n_big : 2 < n no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n n_pos : 0 < n n2_pos : 1 ≤ 2 * n S : Finset ℕ := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1)) f : ℕ → ℕ := fun x => x ^ (Nat.factorization (centralBinom n)) x p : ℕ x✝ : p ∈ Finset.range (2 * n / 3 + 1) h : ¬Nat.Prime p ⊢ p ^ (Nat.factorization (centralBinom n)) p = 1

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals norm_num1 · apply rpow_pos_of_pos four_pos #align bertrand.real_main_inequality Bertrand.real_main_inequality end Bertrand end Real section Nat open Nat /-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast] refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large)) gcongr · have n2_pos : 0 < 2 * n := by positivity exact mod_cast n2_pos · exact_mod_cast Real.nat_sqrt_le_real_sqrt · norm_num1 · exact cast_div_le.trans (by norm_cast) #align bertrand_main_inequality bertrand_main_inequality /-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm apply Finset.prod_subset · exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _) intro x hx h2x rw [Finset.mem_range, lt_succ_iff] at hx h2x rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x replace no_prime := not_exists.mp no_prime x rw [← and_assoc, not_and', not_and_or, not_lt] at no_prime cases' no_prime hx with h h · rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero] · rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero] #align central_binom_factorization_small centralBinom_factorization_small /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate. The bound splits the prime factors of `centralBinom n` into those 1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime. 2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total. 3. Between `2 * n / 3` and `n`, which do not exist. 4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false. 5. Above `2 * n`, which do not exist. -/ theorem centralBinom_le_of_no_bertrand_prime (n : ℕ) (n_big : 2 < n) (no_prime : ¬∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n) : centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big have n2_pos : 1 ≤ 2 * n := mul_pos (zero_lt_two' ℕ) n_pos let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime let f x := x ^ n.centralBinom.factorization x have : ∏ x : ℕ in S, f x = ∏ x : ℕ in Finset.range (2 * n / 3 + 1), f x := by refine' Finset.prod_filter_of_ne fun p _ h => _ contrapose! h; dsimp only

rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]

/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate. The bound splits the prime factors of `centralBinom n` into those 1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime. 2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total. 3. Between `2 * n / 3` and `n`, which do not exist. 4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false. 5. Above `2 * n`, which do not exist. -/ theorem centralBinom_le_of_no_bertrand_prime (n : ℕ) (n_big : 2 < n) (no_prime : ¬∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n) : centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big have n2_pos : 1 ≤ 2 * n := mul_pos (zero_lt_two' ℕ) n_pos let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime let f x := x ^ n.centralBinom.factorization x have : ∏ x : ℕ in S, f x = ∏ x : ℕ in Finset.range (2 * n / 3 + 1), f x := by refine' Finset.prod_filter_of_ne fun p _ h => _ contrapose! h; dsimp only

Mathlib.NumberTheory.Bertrand.155_0.gJXoOT9Ce2wC0xc

/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate. The bound splits the prime factors of `centralBinom n` into those 1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime. 2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total. 3. Between `2 * n / 3` and `n`, which do not exist. 4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false. 5. Above `2 * n`, which do not exist. -/ theorem centralBinom_le_of_no_bertrand_prime (n : ℕ) (n_big : 2 < n) (no_prime : ¬∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n) : centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3)

Mathlib_NumberTheory_Bertrand

n : ℕ n_big : 2 < n no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n n_pos : 0 < n n2_pos : 1 ≤ 2 * n S : Finset ℕ := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1)) f : ℕ → ℕ := fun x => x ^ (Nat.factorization (centralBinom n)) x this : ∏ x in S, f x = ∏ x in Finset.range (2 * n / 3 + 1), f x ⊢ centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3)

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals norm_num1 · apply rpow_pos_of_pos four_pos #align bertrand.real_main_inequality Bertrand.real_main_inequality end Bertrand end Real section Nat open Nat /-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast] refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large)) gcongr · have n2_pos : 0 < 2 * n := by positivity exact mod_cast n2_pos · exact_mod_cast Real.nat_sqrt_le_real_sqrt · norm_num1 · exact cast_div_le.trans (by norm_cast) #align bertrand_main_inequality bertrand_main_inequality /-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm apply Finset.prod_subset · exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _) intro x hx h2x rw [Finset.mem_range, lt_succ_iff] at hx h2x rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x replace no_prime := not_exists.mp no_prime x rw [← and_assoc, not_and', not_and_or, not_lt] at no_prime cases' no_prime hx with h h · rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero] · rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero] #align central_binom_factorization_small centralBinom_factorization_small /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate. The bound splits the prime factors of `centralBinom n` into those 1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime. 2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total. 3. Between `2 * n / 3` and `n`, which do not exist. 4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false. 5. Above `2 * n`, which do not exist. -/ theorem centralBinom_le_of_no_bertrand_prime (n : ℕ) (n_big : 2 < n) (no_prime : ¬∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n) : centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big have n2_pos : 1 ≤ 2 * n := mul_pos (zero_lt_two' ℕ) n_pos let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime let f x := x ^ n.centralBinom.factorization x have : ∏ x : ℕ in S, f x = ∏ x : ℕ in Finset.range (2 * n / 3 + 1), f x := by refine' Finset.prod_filter_of_ne fun p _ h => _ contrapose! h; dsimp only rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]

rw [centralBinom_factorization_small n n_big no_prime, ← this, ← Finset.prod_filter_mul_prod_filter_not S (· ≤ sqrt (2 * n))]

/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate. The bound splits the prime factors of `centralBinom n` into those 1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime. 2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total. 3. Between `2 * n / 3` and `n`, which do not exist. 4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false. 5. Above `2 * n`, which do not exist. -/ theorem centralBinom_le_of_no_bertrand_prime (n : ℕ) (n_big : 2 < n) (no_prime : ¬∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n) : centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big have n2_pos : 1 ≤ 2 * n := mul_pos (zero_lt_two' ℕ) n_pos let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime let f x := x ^ n.centralBinom.factorization x have : ∏ x : ℕ in S, f x = ∏ x : ℕ in Finset.range (2 * n / 3 + 1), f x := by refine' Finset.prod_filter_of_ne fun p _ h => _ contrapose! h; dsimp only rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]

Mathlib.NumberTheory.Bertrand.155_0.gJXoOT9Ce2wC0xc

/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate. The bound splits the prime factors of `centralBinom n` into those 1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime. 2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total. 3. Between `2 * n / 3` and `n`, which do not exist. 4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false. 5. Above `2 * n`, which do not exist. -/ theorem centralBinom_le_of_no_bertrand_prime (n : ℕ) (n_big : 2 < n) (no_prime : ¬∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n) : centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3)

Mathlib_NumberTheory_Bertrand

n : ℕ n_big : 2 < n no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n n_pos : 0 < n n2_pos : 1 ≤ 2 * n S : Finset ℕ := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1)) f : ℕ → ℕ := fun x => x ^ (Nat.factorization (centralBinom n)) x this : ∏ x in S, f x = ∏ x in Finset.range (2 * n / 3 + 1), f x ⊢ (∏ x in Finset.filter (fun x => x ≤ sqrt (2 * n)) S, f x) * ∏ x in Finset.filter (fun x => ¬x ≤ sqrt (2 * n)) S, f x ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3)

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals norm_num1 · apply rpow_pos_of_pos four_pos #align bertrand.real_main_inequality Bertrand.real_main_inequality end Bertrand end Real section Nat open Nat /-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast] refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large)) gcongr · have n2_pos : 0 < 2 * n := by positivity exact mod_cast n2_pos · exact_mod_cast Real.nat_sqrt_le_real_sqrt · norm_num1 · exact cast_div_le.trans (by norm_cast) #align bertrand_main_inequality bertrand_main_inequality /-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm apply Finset.prod_subset · exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _) intro x hx h2x rw [Finset.mem_range, lt_succ_iff] at hx h2x rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x replace no_prime := not_exists.mp no_prime x rw [← and_assoc, not_and', not_and_or, not_lt] at no_prime cases' no_prime hx with h h · rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero] · rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero] #align central_binom_factorization_small centralBinom_factorization_small /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate. The bound splits the prime factors of `centralBinom n` into those 1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime. 2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total. 3. Between `2 * n / 3` and `n`, which do not exist. 4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false. 5. Above `2 * n`, which do not exist. -/ theorem centralBinom_le_of_no_bertrand_prime (n : ℕ) (n_big : 2 < n) (no_prime : ¬∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n) : centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big have n2_pos : 1 ≤ 2 * n := mul_pos (zero_lt_two' ℕ) n_pos let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime let f x := x ^ n.centralBinom.factorization x have : ∏ x : ℕ in S, f x = ∏ x : ℕ in Finset.range (2 * n / 3 + 1), f x := by refine' Finset.prod_filter_of_ne fun p _ h => _ contrapose! h; dsimp only rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero] rw [centralBinom_factorization_small n n_big no_prime, ← this, ← Finset.prod_filter_mul_prod_filter_not S (· ≤ sqrt (2 * n))]

apply mul_le_mul'

/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate. The bound splits the prime factors of `centralBinom n` into those 1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime. 2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total. 3. Between `2 * n / 3` and `n`, which do not exist. 4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false. 5. Above `2 * n`, which do not exist. -/ theorem centralBinom_le_of_no_bertrand_prime (n : ℕ) (n_big : 2 < n) (no_prime : ¬∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n) : centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big have n2_pos : 1 ≤ 2 * n := mul_pos (zero_lt_two' ℕ) n_pos let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime let f x := x ^ n.centralBinom.factorization x have : ∏ x : ℕ in S, f x = ∏ x : ℕ in Finset.range (2 * n / 3 + 1), f x := by refine' Finset.prod_filter_of_ne fun p _ h => _ contrapose! h; dsimp only rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero] rw [centralBinom_factorization_small n n_big no_prime, ← this, ← Finset.prod_filter_mul_prod_filter_not S (· ≤ sqrt (2 * n))]

Mathlib.NumberTheory.Bertrand.155_0.gJXoOT9Ce2wC0xc

/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate. The bound splits the prime factors of `centralBinom n` into those 1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime. 2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total. 3. Between `2 * n / 3` and `n`, which do not exist. 4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false. 5. Above `2 * n`, which do not exist. -/ theorem centralBinom_le_of_no_bertrand_prime (n : ℕ) (n_big : 2 < n) (no_prime : ¬∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n) : centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3)

Mathlib_NumberTheory_Bertrand

case h₁ n : ℕ n_big : 2 < n no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n n_pos : 0 < n n2_pos : 1 ≤ 2 * n S : Finset ℕ := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1)) f : ℕ → ℕ := fun x => x ^ (Nat.factorization (centralBinom n)) x this : ∏ x in S, f x = ∏ x in Finset.range (2 * n / 3 + 1), f x ⊢ ∏ x in Finset.filter (fun x => x ≤ sqrt (2 * n)) S, f x ≤ (2 * n) ^ sqrt (2 * n)

/- Copyright (c) 2020 Patrick Stevens. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Stevens, Bolton Bailey -/ import Mathlib.Data.Nat.Choose.Factorization import Mathlib.Data.Nat.PrimeNormNum import Mathlib.NumberTheory.Primorial import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.Convex.SpecificFunctions.Deriv #align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39" /-! # Bertrand's Postulate This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` an explicit list of primes is provided which covers the remaining cases. As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from [Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`. ## References * [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] * [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] * [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] ## Tags Bertrand, prime, binomial coefficients -/ open scoped BigOperators section Real open Real namespace Bertrand /-- A reified version of the `Bertrand.main_inequality` below. This is not best possible: it actually holds for 464 ≤ x. -/ theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h => div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _) have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by intro x h5 have h6 := mul_pos (zero_lt_two' ℝ) h5 have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)) rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5] -- porting note: the proof was rewritten, because it was too slow have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by apply ConcaveOn.sub apply ConcaveOn.add exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5) convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : ℝ) • LinearMap.id))) using 1 · ext x simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul] rw [← mul_lt_mul_left (two_pos)] norm_num1 rfl apply ConvexOn.smul refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1) exact convexOn_id (convex_Ioi (0.5 : ℝ)) suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩ · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonneg_iff, this] rw [one_le_div] <;> norm_num1 apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1 apply Real.rpow_le_rpow <;> norm_num1 apply rpow_nonneg_of_nonneg; norm_num1 apply rpow_pos_of_pos; norm_num1 apply hf' 18; norm_num1 norm_num1 · have : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1 · conv in 512 => equals 2 ^ 9 => norm_num1 conv in 1024 => equals 2 ^ 10 => norm_num1 conv in 32 => rw [← Nat.cast_ofNat] rw [rpow_nat_cast, ← pow_mul, ← pow_add] conv in 4 => equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 rw [← rpow_mul, ← rpow_nat_cast] apply rpow_le_rpow_of_exponent_le all_goals norm_num1 · apply rpow_pos_of_pos four_pos #align bertrand.real_main_inequality Bertrand.real_main_inequality end Bertrand end Real section Nat open Nat /-- The inequality which contradicts Bertrand's postulate, for large enough `n`. -/ theorem bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := by rw [← @cast_le ℝ] simp only [cast_add, cast_one, cast_mul, cast_pow, ← Real.rpow_nat_cast] refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large)) gcongr · have n2_pos : 0 < 2 * n := by positivity exact mod_cast n2_pos · exact_mod_cast Real.nat_sqrt_le_real_sqrt · norm_num1 · exact cast_div_le.trans (by norm_cast) #align bertrand_main_inequality bertrand_main_inequality /-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. -/ theorem centralBinom_factorization_small (n : ℕ) (n_large : 2 < n) (no_prime : ¬∃ p : ℕ, p.Prime ∧ n < p ∧ p ≤ 2 * n) : centralBinom n = ∏ p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm apply Finset.prod_subset · exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _) intro x hx h2x rw [Finset.mem_range, lt_succ_iff] at hx h2x rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x replace no_prime := not_exists.mp no_prime x rw [← and_assoc, not_and', not_and_or, not_lt] at no_prime cases' no_prime hx with h h · rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero] · rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero] #align central_binom_factorization_small centralBinom_factorization_small /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate. The bound splits the prime factors of `centralBinom n` into those 1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime. 2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total. 3. Between `2 * n / 3` and `n`, which do not exist. 4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false. 5. Above `2 * n`, which do not exist. -/ theorem centralBinom_le_of_no_bertrand_prime (n : ℕ) (n_big : 2 < n) (no_prime : ¬∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n) : centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big have n2_pos : 1 ≤ 2 * n := mul_pos (zero_lt_two' ℕ) n_pos let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime let f x := x ^ n.centralBinom.factorization x have : ∏ x : ℕ in S, f x = ∏ x : ℕ in Finset.range (2 * n / 3 + 1), f x := by refine' Finset.prod_filter_of_ne fun p _ h => _ contrapose! h; dsimp only rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero] rw [centralBinom_factorization_small n n_big no_prime, ← this, ← Finset.prod_filter_mul_prod_filter_not S (· ≤ sqrt (2 * n))] apply mul_le_mul' ·

refine' (Finset.prod_le_prod' fun p _ => (_ : f p ≤ 2 * n)).trans _

/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate. The bound splits the prime factors of `centralBinom n` into those 1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime. 2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total. 3. Between `2 * n / 3` and `n`, which do not exist. 4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false. 5. Above `2 * n`, which do not exist. -/ theorem centralBinom_le_of_no_bertrand_prime (n : ℕ) (n_big : 2 < n) (no_prime : ¬∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n) : centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big have n2_pos : 1 ≤ 2 * n := mul_pos (zero_lt_two' ℕ) n_pos let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime let f x := x ^ n.centralBinom.factorization x have : ∏ x : ℕ in S, f x = ∏ x : ℕ in Finset.range (2 * n / 3 + 1), f x := by refine' Finset.prod_filter_of_ne fun p _ h => _ contrapose! h; dsimp only rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero] rw [centralBinom_factorization_small n n_big no_prime, ← this, ← Finset.prod_filter_mul_prod_filter_not S (· ≤ sqrt (2 * n))] apply mul_le_mul' ·

Mathlib.NumberTheory.Bertrand.155_0.gJXoOT9Ce2wC0xc

/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate. The bound splits the prime factors of `centralBinom n` into those 1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime. 2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total. 3. Between `2 * n / 3` and `n`, which do not exist. 4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false. 5. Above `2 * n`, which do not exist. -/ theorem centralBinom_le_of_no_bertrand_prime (n : ℕ) (n_big : 2 < n) (no_prime : ¬∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n) : centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3)

Mathlib_NumberTheory_Bertrand