| import Mathlib | |
| set_option linter.unusedVariables.analyzeTactics true | |
| open Nat BigOperators Finset | |
| lemma aux_1 | |
| (a : ℕ) : | |
| ¬ a ^ 2 ≡ 2 [MOD 5] := by | |
| intro ha₀ | |
| induction' a with n hn | |
| . simp at ha₀ | |
| have ha₁: ¬ 0 ≡ 2 [MOD 5] := by decide | |
| exact ha₁ ha₀ | |
| . let b:ℕ := n % 5 | |
| have hb₀: b < 5 := by omega | |
| have hb₁: n ≡ b [MOD 5] := by exact Nat.ModEq.symm (Nat.mod_modEq n 5) | |
| have hb₂: (n + 1) ≡ (b + 1) [MOD 5] := by | |
| exact Nat.ModEq.add_right 1 hb₁ | |
| have hb₃: (n + 1) ^ 2 ≡ (b + 1) ^ 2 [MOD 5] := by | |
| exact Nat.ModEq.pow 2 hb₂ | |
| interval_cases b | |
| . simp at * | |
| have g₀: 1 ≡ 2 [MOD 5] := by | |
| refine Nat.ModEq.trans hb₃.symm ha₀ | |
| have g₁: ¬ 1 ≡ 2 [MOD 5] := by decide | |
| exact g₁ g₀ | |
| . simp at hb₃ | |
| have g₀: 4 ≡ 2 [MOD 5] := by | |
| refine Nat.ModEq.trans hb₃.symm ha₀ | |
| have g₁: ¬ 4 ≡ 2 [MOD 5] := by decide | |
| exact g₁ g₀ | |
| . simp at hb₃ | |
| have g₀: 9 ≡ 2 [MOD 5] := by | |
| refine Nat.ModEq.trans hb₃.symm ha₀ | |
| have g₁: ¬ 9 ≡ 2 [MOD 5] := by decide | |
| exact g₁ g₀ | |
| . simp at hb₃ | |
| have g₀: 16 ≡ 2 [MOD 5] := by | |
| refine Nat.ModEq.trans hb₃.symm ha₀ | |
| have g₁: ¬ 16 ≡ 2 [MOD 5] := by decide | |
| exact g₁ g₀ | |
| . simp at hb₃ | |
| have g₀: 25 ≡ 2 [MOD 5] := by | |
| refine Nat.ModEq.trans hb₃.symm ha₀ | |
| have g₁: ¬ 25 ≡ 2 [MOD 5] := by decide | |
| exact g₁ g₀ | |
| lemma aux_2 | |
| (a : ℕ) : | |
| ¬ a ^ 2 ≡ 3 [MOD 5] := by | |
| intro ha₀ | |
| induction' a with n hn | |
| . simp at ha₀ | |
| have ha₁: ¬ 0 ≡ 3 [MOD 5] := by decide | |
| exact ha₁ ha₀ | |
| . let b:ℕ := n % 5 | |
| have hb₀: b < 5 := by omega | |
| have hb₁: n ≡ b [MOD 5] := by exact Nat.ModEq.symm (Nat.mod_modEq n 5) | |
| have hb₂: (n + 1) ≡ (b + 1) [MOD 5] := by | |
| exact Nat.ModEq.add_right 1 hb₁ | |
| have hb₃: (n + 1) ^ 2 ≡ (b + 1) ^ 2 [MOD 5] := by | |
| exact Nat.ModEq.pow 2 hb₂ | |
| interval_cases b | |
| . simp at * | |
| have g₀: 1 ≡ 3 [MOD 5] := by | |
| refine Nat.ModEq.trans hb₃.symm ha₀ | |
| have g₁: ¬ 1 ≡ 3 [MOD 5] := by decide | |
| exact g₁ g₀ | |
| . simp at hb₃ | |
| have g₀: 4 ≡ 3 [MOD 5] := by | |
| refine Nat.ModEq.trans hb₃.symm ha₀ | |
| have g₁: ¬ 4 ≡ 3 [MOD 5] := by decide | |
| exact g₁ g₀ | |
| . simp at hb₃ | |
| have g₀: 9 ≡ 3 [MOD 5] := by | |
| refine Nat.ModEq.trans hb₃.symm ha₀ | |
| have g₁: ¬ 9 ≡ 3 [MOD 5] := by decide | |
| exact g₁ g₀ | |
| . simp at hb₃ | |
| have g₀: 16 ≡ 3 [MOD 5] := by | |
| refine Nat.ModEq.trans hb₃.symm ha₀ | |
| have g₁: ¬ 16 ≡ 3 [MOD 5] := by decide | |
| exact g₁ g₀ | |
| . simp at hb₃ | |
| have g₀: 25 ≡ 3 [MOD 5] := by | |
| refine Nat.ModEq.trans hb₃.symm ha₀ | |
| have g₁: ¬ 25 ≡ 3 [MOD 5] := by decide | |
| exact g₁ g₀ | |
| lemma aux_3 | |
| (n : ℕ) : | |
| 7 ^ (2 * n + 1) ≡ 2 [MOD 5] ∨ 7 ^ (2 * n + 1) ≡ 3 [MOD 5] := by | |
| induction' n with d hd | |
| . simp | |
| left | |
| decide | |
| . let b:ℕ := (7 ^ (2 * d + 1)) % 5 | |
| have hb: b = (7 ^ (2 * d + 1)) % 5 := by rfl | |
| have hb₀: b < 5 := by | |
| rw [hb] | |
| omega | |
| have hb₁: (7 ^ (2 * d + 1)) ≡ b [MOD 5] := by | |
| exact ModEq.symm (mod_modEq (7 ^ (2 * d + 1)) 5) | |
| ring_nf at * | |
| have hb₂: 7 ^ (d * 2) * 7 * 49 ≡ b * 49 [MOD 5] := by | |
| exact ModEq.mul hb₁ rfl | |
| have hb₃: 7 ^ (d * 2) * 7 * 49 ≡ 2 * 49 [MOD 5] ∨ 7 ^ (d * 2) * 7 * 49 ≡ 3 * 49 [MOD 5] := by | |
| cases' hd with hd₀ hd₁ | |
| . left | |
| exact ModEq.mul hd₀ rfl | |
| . right | |
| exact ModEq.mul hd₁ rfl | |
| ring_nf at hb₂ | |
| ring_nf at * | |
| cases' hb₃ with hb₄ hb₅ | |
| . interval_cases b | |
| . ring_nf at hb₂ | |
| have g₀: 0 ≡ 98 [MOD 5] := by | |
| refine Nat.ModEq.trans hb₂.symm hb₄ | |
| have g₁: ¬ 0 ≡ 98 [MOD 5] := by decide | |
| exact (g₁ g₀).elim | |
| . ring_nf at hb₂ | |
| have g₀: 49 ≡ 98 [MOD 5] := by | |
| refine Nat.ModEq.trans hb₂.symm hb₄ | |
| have g₁: ¬ 49 ≡ 98 [MOD 5] := by decide | |
| exact (g₁ g₀).elim | |
| . ring_nf at hb₂ | |
| have g₀: 98 ≡ 3 [MOD 5] := by decide | |
| right | |
| refine Nat.ModEq.trans hb₂ g₀ | |
| . ring_nf at hb₂ | |
| have g₀: 147 ≡ 98 [MOD 5] := by | |
| refine Nat.ModEq.trans hb₂.symm hb₄ | |
| have g₁: ¬ 147 ≡ 98 [MOD 5] := by decide | |
| exact (g₁ g₀).elim | |
| . ring_nf at hb₂ | |
| have g₀: 196 ≡ 98 [MOD 5] := by | |
| refine Nat.ModEq.trans hb₂.symm hb₄ | |
| have g₁: ¬ 196 ≡ 98 [MOD 5] := by decide | |
| exact (g₁ g₀).elim | |
| . interval_cases b | |
| . ring_nf at hb₂ | |
| have g₀: 0 ≡ 147 [MOD 5] := by | |
| refine Nat.ModEq.trans hb₂.symm hb₅ | |
| have g₁: ¬ 0 ≡ 147 [MOD 5] := by decide | |
| exact (g₁ g₀).elim | |
| . ring_nf at hb₂ | |
| have g₀: 49 ≡ 147 [MOD 5] := by | |
| refine Nat.ModEq.trans hb₂.symm hb₅ | |
| have g₁: ¬ 49 ≡ 147 [MOD 5] := by decide | |
| exact (g₁ g₀).elim | |
| . ring_nf at hb₂ | |
| have g₀: 98 ≡ 147 [MOD 5] := by | |
| refine Nat.ModEq.trans hb₂.symm hb₅ | |
| have g₁: ¬ 98 ≡ 147 [MOD 5] := by decide | |
| exact (g₁ g₀).elim | |
| . ring_nf at hb₂ | |
| exact Or.intro_left (7 ^ (d * 2) * 343 ≡ 3 [MOD 5]) hb₅ | |
| . ring_nf at hb₂ | |
| have g₀: 196 ≡ 147 [MOD 5] := by | |
| refine Nat.ModEq.trans hb₂.symm hb₅ | |
| have g₁: ¬ 196 ≡ 147 [MOD 5] := by decide | |
| exact (g₁ g₀).elim | |
| lemma aux_4 | |
| (n b a : ℕ) | |
| (k : ℝ) | |
| -- (hk : k = √8) | |
| -- (hb : b = ∑ k ∈ range (n + 1), (2 * n + 1).choose (2 * k + 1) * 2 ^ (3 * k)) | |
| -- (ha : a = ∑ k ∈ range (n + 1), (2 * n + 1).choose (2 * k) * 2 ^ (3 * k)) | |
| (hb₁ : ↑b = 1 / k * ∑ x ∈ range (n + 1), ↑((2 * n + 1).choose (2 * x + 1)) * k ^ (2 * x + 1)) | |
| (ha₁ : ↑a = ∑ x ∈ range (n + 1), ↑((2 * n + 1).choose (2 * x)) * k ^ (2 * x)) | |
| (hk₀ : k * k⁻¹ = 1) : | |
| (1 + k) ^ (2 * n + 1) = ↑a + ↑b * k := by | |
| rw [mul_comm _ k, hb₁, ← mul_assoc] | |
| rw [← inv_eq_one_div, hk₀, one_mul, ha₁] | |
| rw [add_comm, add_pow k 1 (2 * n + 1)] | |
| simp | |
| clear hb₁ ha₁ b a hk₀ | |
| let f : ℕ → ℝ := fun i => ↑((2 * n + 1).choose (i)) * k ^ i | |
| let fs₂ := Finset.range (2 * n + 2) | |
| -- let fs₀ : Finset ℕ := Finset.filter (fun x => Odd x) (Finset.range (2 * n + 2)) | |
| let fs₀ : Finset ℕ := fs₂.filter (fun x => Odd x) | |
| let fs₁ : Finset ℕ := fs₂.filter (fun x => Even x) | |
| let fs₃ : Finset ℕ := Finset.range (n + 1) | |
| have h₀: ∑ x ∈ range (n + 1), ↑((2 * n + 1).choose (2 * x + 1)) * k ^ (2 * x + 1) = | |
| ∑ x ∈ fs₀, ↑((2 * n + 1).choose (x)) * k ^ (x) := by | |
| have h₀₁: ∑ x ∈ fs₃, f (2 * x + 1) = ∑ x ∈ (fs₀), f x := by | |
| refine sum_bij ?i ?_ ?i_inj ?i_surj ?h | |
| . intros a _ | |
| exact (2 * a + 1) | |
| . intros a ha₀ | |
| have ha₁: a ≤ n := by exact mem_range_succ_iff.mp ha₀ | |
| have ha₂: 2 * a + 1 ≤ 2 * n + 1 := by linarith | |
| have ha₃: (2 * a + 1) ∈ fs₂ := by exact mem_range_succ_iff.mpr ha₂ | |
| have ha₄: Odd (2 * a + 1) := by exact odd_two_mul_add_one a | |
| refine mem_filter.mpr ?_ | |
| exact And.symm ⟨ha₄, ha₃⟩ | |
| . intros a _ b _ h₃ | |
| linarith | |
| . intros b hb₀ | |
| use ((b - 1) / 2) | |
| refine exists_prop.mpr ?_ | |
| have hb₁: b ∈ fs₂ ∧ Odd b := by exact mem_filter.mp hb₀ | |
| have hb₂: 1 ≤ b := by | |
| by_contra! hc₀ | |
| interval_cases b | |
| have hc₁: ¬ Odd 0 := by decide | |
| apply hc₁ hb₁.2 | |
| have hb₃: Even (b - 1) := by | |
| refine (Nat.even_sub hb₂).mpr ?_ | |
| simp only [not_even_one, iff_false, not_even_iff_odd] | |
| exact hb₁.2 | |
| constructor | |
| . have hb₄: b < 2 * n + 2 := by exact List.mem_range.mp hb₁.1 | |
| have hb₅: (b - 1) / 2 < n + 1 := by omega | |
| exact mem_range.mpr hb₅ | |
| . have hb₆: 2 * ((b - 1) / 2) = b - 1 := by exact two_mul_div_two_of_even hb₃ | |
| rw [hb₆] | |
| exact Nat.sub_add_cancel hb₂ | |
| . exact fun a _ => rfl | |
| exact h₀₁ | |
| have h₁: ∑ x ∈ range (n + 1), ↑((2 * n + 1).choose (2 * x)) * k ^ (2 * x) = | |
| ∑ x ∈ fs₁, ↑((2 * n + 1).choose (x)) * k ^ (x) := by | |
| have h₁₁: ∑ x ∈ fs₃, f (2 * x) = ∑ x ∈ (fs₁), f x := by | |
| refine sum_bij ?_ ?_ ?_ ?_ ?_ | |
| . intros a _ | |
| exact (2 * a) | |
| . intros a ha₀ | |
| have ha₁: a < n + 1 := by exact List.mem_range.mp ha₀ | |
| have ha₂: 2 * a < 2 * n + 2 := by linarith | |
| refine mem_filter.mpr ?_ | |
| constructor | |
| . exact mem_range.mpr ha₂ | |
| . exact even_two_mul a | |
| . intros a _ b _ h₃ | |
| exact Nat.eq_of_mul_eq_mul_left (by norm_num) h₃ | |
| . intros b hb₀ | |
| use (b/2) | |
| refine exists_prop.mpr ?_ | |
| have hb₁: b ∈ fs₂ ∧ Even b := by exact mem_filter.mp hb₀ | |
| constructor | |
| . have hb₂: b < 2 * n + 2 := by exact List.mem_range.mp hb₁.1 | |
| have hb₃: (b / 2) < n + 1 := by exact Nat.div_lt_of_lt_mul hb₂ | |
| exact mem_range.mpr hb₃ | |
| . exact two_mul_div_two_of_even hb₁.2 | |
| . exact fun a _ => rfl | |
| exact h₁₁ | |
| have h₂: ∑ x ∈ range (2 * n + 1 + 1), k ^ x * ↑((2 * n + 1).choose x) = | |
| ∑ x ∈ fs₂, ↑((2 * n + 1).choose x) * k ^ x := by | |
| refine Finset.sum_congr (rfl) ?_ | |
| intros x _ | |
| rw [mul_comm] | |
| rw [h₀, h₁, h₂] | |
| have h₃: fs₂ = fs₀ ∪ fs₁ := by | |
| refine Finset.ext_iff.mpr ?_ | |
| intro a | |
| constructor | |
| . intro ha₀ | |
| refine mem_union.mpr ?mp.a | |
| have ha₁: Odd a ∨ Even a := by exact Or.symm (even_or_odd a) | |
| cases' ha₁ with ha₂ ha₃ | |
| . left | |
| refine mem_filter.mpr ?mp.a.inl.h.a | |
| exact And.symm ⟨ha₂, ha₀⟩ | |
| . right | |
| refine mem_filter.mpr ?mp.a.inl.h.b | |
| exact And.symm ⟨ha₃, ha₀⟩ | |
| . intro ha₀ | |
| apply mem_union.mp at ha₀ | |
| cases' ha₀ with ha₁ ha₂ | |
| . exact mem_of_mem_filter a ha₁ | |
| . exact mem_of_mem_filter a ha₂ | |
| have h₄: Disjoint fs₀ fs₁ := by | |
| refine disjoint_filter.mpr ?_ | |
| intros x _ hx₁ | |
| exact not_even_iff_odd.mpr hx₁ | |
| nth_rw 2 [add_comm] | |
| rw [h₃, Finset.sum_union h₄] | |
| lemma aux_5 | |
| (n b a : ℕ) | |
| (k : ℝ) | |
| -- (hk : k = √8) | |
| -- (hb : b = ∑ k ∈ range (n + 1), (2 * n + 1).choose (2 * k + 1) * 2 ^ (3 * k)) | |
| -- (ha : a = ∑ k ∈ range (n + 1), (2 * n + 1).choose (2 * k) * 2 ^ (3 * k)) | |
| (hb₁ : ↑b = 1 / k * ∑ x ∈ range (n + 1), ↑((2 * n + 1).choose (2 * x + 1)) * k ^ (2 * x + 1)) | |
| (ha₁ : ↑a = ∑ x ∈ range (n + 1), ↑((2 * n + 1).choose (2 * x)) * k ^ (2 * x)) | |
| (hk₀ : k * k⁻¹ = 1) : | |
| (1 - k) ^ (2 * n + 1) = ↑a - ↑b * k := by | |
| rw [mul_comm _ k, hb₁, ← mul_assoc] | |
| rw [← inv_eq_one_div, hk₀, one_mul, ha₁, sub_eq_add_neg] | |
| rw [add_comm 1 _, add_pow (-k) 1 (2 * n + 1)] | |
| simp | |
| clear hb₁ ha₁ b a hk₀ | |
| let f₀ : ℕ → ℝ := fun i => ↑((2 * n + 1).choose (i)) * k ^ i | |
| let f₁ : ℕ → ℝ := fun i => ↑((2 * n + 1).choose (i)) * (-k) ^ i | |
| let fs₂ := Finset.range (2 * n + 2) | |
| let fs₀ : Finset ℕ := fs₂.filter (fun x => Odd x) | |
| let fs₁ : Finset ℕ := fs₂.filter (fun x => Even x) | |
| let fs₃ : Finset ℕ := Finset.range (n + 1) | |
| have h₀: ∑ x ∈ range (n + 1), ↑((2 * n + 1).choose (2 * x + 1)) * k ^ (2 * x + 1) = | |
| - ∑ x ∈ fs₀, ↑((2 * n + 1).choose (x)) * (-k) ^ (x) := by | |
| rw [neg_eq_neg_one_mul, Finset.mul_sum] | |
| have h₀₁: ∑ x ∈ fs₃, f₀ (2 * x + 1) = ∑ x ∈ (fs₀), -1 * f₁ x := by | |
| refine sum_bij ?i ?_ ?i_inj ?i_surj ?h | |
| . intros a _ | |
| exact (2 * a + 1) | |
| . intros a ha₀ | |
| have ha₁: a ≤ n := by exact mem_range_succ_iff.mp ha₀ | |
| have ha₂: 2 * a + 1 ≤ 2 * n + 1 := by linarith | |
| have ha₃: (2 * a + 1) ∈ fs₂ := by exact mem_range_succ_iff.mpr ha₂ | |
| have ha₄: Odd (2 * a + 1) := by exact odd_two_mul_add_one a | |
| refine mem_filter.mpr ?_ | |
| exact And.symm ⟨ha₄, ha₃⟩ | |
| . intros a _ b _ h₃ | |
| linarith | |
| . intros b hb₀ | |
| use ((b - 1) / 2) | |
| refine exists_prop.mpr ?_ | |
| have hb₁: b ∈ fs₂ ∧ Odd b := by exact mem_filter.mp hb₀ | |
| have hb₂: 1 ≤ b := by | |
| by_contra! hc₀ | |
| interval_cases b | |
| have hc₁: ¬ Odd 0 := by decide | |
| apply hc₁ hb₁.2 | |
| have hb₃: Even (b - 1) := by | |
| refine (Nat.even_sub hb₂).mpr ?_ | |
| simp only [not_even_one, iff_false, not_even_iff_odd] | |
| exact hb₁.2 | |
| constructor | |
| . have hb₄: b < 2 * n + 2 := by exact List.mem_range.mp hb₁.1 | |
| have hb₅: (b - 1) / 2 < n + 1 := by omega | |
| exact mem_range.mpr hb₅ | |
| . have hb₆: 2 * ((b - 1) / 2) = b - 1 := by exact two_mul_div_two_of_even hb₃ | |
| rw [hb₆] | |
| exact Nat.sub_add_cancel hb₂ | |
| . intros b hb₀ | |
| ring_nf | |
| have hb₁: (-1:ℝ) ^ (b * 2) = 1 := by | |
| refine (neg_one_pow_eq_one_iff_even (by norm_num)).mpr ?_ | |
| rw [mul_comm] | |
| exact even_two_mul b | |
| rw [hb₁, mul_one] | |
| exact h₀₁ | |
| have h₁: ∑ x ∈ range (n + 1), ↑((2 * n + 1).choose (2 * x)) * k ^ (2 * x) = | |
| ∑ x ∈ fs₁, ↑((2 * n + 1).choose (x)) * (-k) ^ (x) := by | |
| have h₁₁: ∑ x ∈ fs₃, f₀ (2 * x) = ∑ x ∈ (fs₁), f₁ x := by | |
| refine sum_bij ?_ ?_ ?_ ?_ ?_ | |
| . intros a _ | |
| exact (2 * a) | |
| . intros a ha₀ | |
| have ha₁: a < n + 1 := by exact List.mem_range.mp ha₀ | |
| have ha₂: 2 * a < 2 * n + 2 := by linarith | |
| refine mem_filter.mpr ?_ | |
| constructor | |
| . exact mem_range.mpr ha₂ | |
| . exact even_two_mul a | |
| . intros a _ b _ h₃ | |
| exact Nat.eq_of_mul_eq_mul_left (by norm_num) h₃ | |
| . intros b hb₀ | |
| use (b/2) | |
| refine exists_prop.mpr ?_ | |
| have hb₁: b ∈ fs₂ ∧ Even b := by exact mem_filter.mp hb₀ | |
| constructor | |
| . have hb₂: b < 2 * n + 2 := by exact List.mem_range.mp hb₁.1 | |
| have hb₃: (b / 2) < n + 1 := by exact Nat.div_lt_of_lt_mul hb₂ | |
| exact mem_range.mpr hb₃ | |
| . exact two_mul_div_two_of_even hb₁.2 | |
| . intros b hb₀ | |
| ring_nf | |
| have hb₁: (-1:ℝ) ^ (b * 2) = 1 := by | |
| refine (neg_one_pow_eq_one_iff_even (by norm_num)).mpr ?_ | |
| rw [mul_comm] | |
| exact even_two_mul b | |
| rw [hb₁, mul_one] | |
| exact h₁₁ | |
| have h₂: ∑ x ∈ range (2 * n + 1 + 1), (-k) ^ x * ↑((2 * n + 1).choose x) = | |
| ∑ x ∈ fs₂, ↑((2 * n + 1).choose x) * (-k) ^ x := by | |
| refine Finset.sum_congr (rfl) ?_ | |
| intros x _ | |
| rw [mul_comm] | |
| rw [h₀, h₁, h₂, sub_neg_eq_add] | |
| have h₃: fs₂ = fs₀ ∪ fs₁ := by | |
| refine Finset.ext_iff.mpr ?_ | |
| intro a | |
| constructor | |
| . intro ha₀ | |
| refine mem_union.mpr ?mp.a | |
| have ha₁: Odd a ∨ Even a := by exact Or.symm (even_or_odd a) | |
| cases' ha₁ with ha₂ ha₃ | |
| . left | |
| refine mem_filter.mpr ?mp.a.inl.h.a | |
| exact And.symm ⟨ha₂, ha₀⟩ | |
| . right | |
| refine mem_filter.mpr ?mp.a.inl.h.b | |
| exact And.symm ⟨ha₃, ha₀⟩ | |
| . intro ha₀ | |
| apply mem_union.mp at ha₀ | |
| cases' ha₀ with ha₁ ha₂ | |
| . exact mem_of_mem_filter a ha₁ | |
| . exact mem_of_mem_filter a ha₂ | |
| have h₄: Disjoint fs₀ fs₁ := by | |
| refine disjoint_filter.mpr ?_ | |
| intros x _ hx₁ | |
| exact not_even_iff_odd.mpr hx₁ | |
| nth_rw 2 [add_comm] | |
| rw [h₃, Finset.sum_union h₄] | |
| theorem imo_1974_p3 | |
| (n : ℕ) : | |
| ¬ 5 ∣ ∑ k ∈ Finset.range (n + 1), (Nat.choose (2 * n + 1) (2 * k + 1)) * (2^(3 * k)) := by | |
| let k:ℝ := Real.sqrt (8:ℝ) | |
| have hk: k = Real.sqrt (8:ℝ) := by rfl | |
| let b:ℕ := ∑ k ∈ Finset.range (n + 1), (Nat.choose (2 * n + 1) (2 * k + 1)) * (2^(3 * k)) | |
| have hb: b = ∑ k ∈ Finset.range (n + 1), (Nat.choose (2 * n + 1) (2 * k + 1)) * (2^(3 * k)) := by rfl | |
| rw [← hb] | |
| let a:ℕ := ∑ k ∈ Finset.range (n + 1), (Nat.choose (2 * n + 1) (2 * k) * (2 ^ (3 * k))) | |
| have ha: a = ∑ k ∈ Finset.range (n + 1), (Nat.choose (2 * n + 1) (2 * k) * (2 ^ (3 * k))) := by rfl | |
| have hb₁: b = (1 / k) * | |
| ∑ x ∈ Finset.range (n + 1), (Nat.choose (2 * n + 1) (2 * x + 1)) * (k ^ (2 * x + 1)) := by | |
| rw [hb, hk] | |
| simp | |
| rw [Finset.mul_sum] | |
| refine Finset.sum_congr (rfl) ?_ | |
| intros x _ | |
| rw [mul_comm ((√8)⁻¹), mul_assoc] | |
| refine mul_eq_mul_left_iff.mpr ?_ | |
| left | |
| rw [pow_succ, pow_mul, pow_mul, Real.sq_sqrt (by norm_num)] | |
| norm_num | |
| have ha₁: a = ∑ x ∈ Finset.range (n + 1), (Nat.choose (2 * n + 1) (2 * x) * (k ^ (2 * x))) := by | |
| rw [ha, hk] | |
| simp | |
| refine Finset.sum_congr (rfl) ?_ | |
| intros x _ | |
| refine mul_eq_mul_left_iff.mpr ?_ | |
| left | |
| rw [pow_mul, pow_mul, Real.sq_sqrt (by norm_num)] | |
| norm_num | |
| have hk₀: k * k⁻¹ = 1 := by | |
| refine (mul_inv_eq_one₀ ?_).mpr (rfl) | |
| rw [hk] | |
| norm_num | |
| have h₀: (1 + k) ^ (2 * n + 1) = a + b * k := by | |
| exact aux_4 n b a k hb₁ ha₁ hk₀ | |
| have h₁: (1 - k) ^ (2 * n + 1) = a - b * k := by | |
| exact aux_5 n b a k hb₁ ha₁ hk₀ | |
| have h₂: ((1 + k) * (1 - k)) ^ (2 * n + 1) = (a + b * k) * (a - b * k) := by | |
| rw [mul_pow, h₀, h₁] | |
| rw [← sq_sub_sq 1 k] at h₂ | |
| rw [← sq_sub_sq (↑a) ((↑b:ℝ) * k)] at h₂ | |
| rw [mul_pow, hk] at h₂ | |
| norm_num at h₂ | |
| have h₃: (7:ℕ) ^ (2 * n + 1) = b ^ 2 * 8 - a ^ 2 := by | |
| have h₃₀: Odd (2 * n + 1) := by exact odd_two_mul_add_one n | |
| have h₃₁: (-7:ℝ) = (-1:ℝ) * (7:ℕ) := by norm_num | |
| have h₃₂: (-1:ℝ) ^ (2 * n + 1) = -1 := by exact Odd.neg_one_pow h₃₀ | |
| have h₃₃: ↑a ^ 2 - ↑b ^ 2 * 8 = (-1:ℝ) * (↑b ^ 2 * 8 - ↑a ^ 2) := by | |
| linarith | |
| rw [h₃₁, mul_pow, h₃₂, h₃₃] at h₂ | |
| simp at h₂ | |
| have h₃₄: (7:ℝ) ^ (2 * n + 1) = ↑b ^ 2 * 8 - ↑a ^ 2 := by | |
| linarith | |
| norm_cast at h₃₄ | |
| rw [Int.subNatNat_eq_coe] at h₃₄ | |
| rw [← Int.toNat_sub, ← h₃₄] | |
| exact rfl | |
| have h₄: 7 ^ (2 * n + 1) ≡ 2 [MOD 5] ∨ 7 ^ (2 * n + 1) ≡ 3 [MOD 5] := by | |
| refine aux_3 n | |
| by_contra! hc₀ | |
| have hc₁: b^2 * 8 ≡ 0^2 * 8 [MOD 5] := by | |
| refine ModEq.mul ?_ rfl | |
| refine ModEq.pow 2 ?_ | |
| exact modEq_zero_iff_dvd.mpr hc₀ | |
| simp at hc₁ | |
| have h₅: a ^ 2 < b ^ 2 * 8 := by | |
| have h₅₀: 0 < 7 ^ (2 * n + 1) := by | |
| exact Nat.pow_pos (by norm_num) | |
| rw [h₃] at h₅₀ | |
| exact Nat.lt_of_sub_pos h₅₀ | |
| cases' h₄ with h₄₀ h₄₁ | |
| . rw [h₃] at h₄₀ | |
| have hc₂: b ^ 2 * 8 - a ^ 2 + a ^ 2 ≡ 2 + a ^ 2 [MOD 5] := by | |
| exact ModEq.add_right (a ^ 2) h₄₀ | |
| rw [Nat.sub_add_cancel (le_of_lt h₅)] at hc₂ | |
| have hc₃: 3 + (2 + a ^ 2) ≡ 3 [MOD 5] := by | |
| apply Nat.ModEq.trans hc₂.symm at hc₁ | |
| exact ModEq.add_left 3 hc₁ | |
| have hc₄: a ^ 2 ≡ 3 [MOD 5] := by | |
| rw [← add_assoc, ← zero_add 3] at hc₃ | |
| norm_num at hc₃ | |
| have hc₄: 5 ≡ 0 [MOD 5] := by decide | |
| exact Nat.ModEq.add_left_cancel hc₄ hc₃ | |
| have hc₅: ¬ a ^ 2 ≡ 3 [MOD 5] := by exact aux_2 a | |
| exact hc₅ hc₄ | |
| . rw [h₃] at h₄₁ | |
| have hc₂: b ^ 2 * 8 - a ^ 2 + a ^ 2 ≡ 3 + a ^ 2 [MOD 5] := by | |
| exact ModEq.add_right (a ^ 2) h₄₁ | |
| rw [Nat.sub_add_cancel (le_of_lt h₅)] at hc₂ | |
| apply Nat.ModEq.trans hc₂.symm at hc₁ | |
| have hc₃: a ^ 2 ≡ 2 [MOD 5] := by | |
| refine Nat.ModEq.add_left_cancel' 3 ?_ | |
| exact hc₁ | |
| have hc₄: ¬ a ^ 2 ≡ 2 [MOD 5] := by exact aux_1 a | |
| exact hc₄ hc₃ | |