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