import Mathlib set_option linter.unusedVariables.analyzeTactics true open Nat lemma imo_2022_p5_1 (b p: ℕ) (h₀: 0 < b) -- (hp: Nat.prime p) (hbp: b < p) : (1 + (b * p + b ^ p) ≤ (1 + b) ^ p) := by refine Nat.le_induction ?_ ?_ p hbp . rw [add_pow 1 b b.succ] rw [Finset.sum_range_succ _ b.succ] simp rw [add_comm (∑ x ∈ Finset.range (b + 1), b ^ (b + 1 - x) * (b + 1).choose x) 1] simp rw [Finset.sum_range_succ _ b] simp rw [add_comm _ (b * (b + 1))] simp have gb: b = b - 1 + 1 := by exact (Nat.sub_eq_iff_eq_add h₀).mp rfl nth_rewrite 3 [gb] rw [Finset.sum_range_succ' _ (b-1)] simp . intros n _ h₂ nth_rewrite 2 [pow_add] rw [pow_one] have h₃: (1 + (b * n + b ^ n)) * (1 + b) ≤ ((1 + b) ^ n) * (1 + b) := by exact mul_le_mul_right' h₂ (1 + b) have h₄: 1 + (b * (n + 1) + b ^ (n + 1)) ≤ (1 + (b * n + b ^ n)) * (1 + b) := by ring_nf rw [Nat.add_assoc _ (b ^ 2 * n) (b ^ n)] exact Nat.le_add_right (1 + b + b * b ^ n + b * n) (b ^ 2 * n + b ^ n) exact le_trans h₄ h₃ lemma imo_2022_p5_1_1 (b : ℕ) -- (p : ℕ) (h₀ : 0 < b) : -- (hbp : b < p) : 1 + (b * succ b + b ^ succ b) ≤ (1 + b) ^ succ b := by rw [add_pow 1 b b.succ] rw [Finset.sum_range_succ _ b.succ] simp rw [add_comm (∑ x ∈ Finset.range (b + 1), b ^ (b + 1 - x) * (b + 1).choose x) 1] simp rw [Finset.sum_range_succ _ b] simp rw [add_comm _ (b * (b + 1))] simp have gb: b = b - 1 + 1 := by exact (Nat.sub_eq_iff_eq_add h₀).mp rfl nth_rewrite 3 [gb] rw [Finset.sum_range_succ' _ (b-1)] simp lemma imo_2022_p5_1_2 (b : ℕ) -- (p : ℕ) (h₀ : 0 < b) : -- (hbp : b < p) : 1 + (b * succ b + b ^ succ b) ≤ (Finset.sum (Finset.range (succ b)) fun x => 1 ^ x * b ^ (succ b - x) * ↑(choose (succ b) x)) + 1 ^ succ b * b ^ (succ b - succ b) * ↑(choose (succ b) (succ b)) := by simp rw [add_comm (∑ x ∈ Finset.range (b + 1), b ^ (b + 1 - x) * (b + 1).choose x) 1] simp rw [Finset.sum_range_succ _ b] simp rw [add_comm _ (b * (b + 1))] simp have gb: b = b - 1 + 1 := by exact (Nat.sub_eq_iff_eq_add h₀).mp rfl nth_rewrite 3 [gb] rw [Finset.sum_range_succ' _ (b-1)] simp lemma imo_2022_p5_1_3 (b : ℕ) -- (p : ℕ) -- (h₀ : 0 < b) -- (hbp : b < p) (h₁ : 1 + (b * succ b + b ^ succ b) ≤ (Finset.sum (Finset.range (succ b)) fun x => 1 ^ x * b ^ (succ b - x) * ↑(choose (succ b) x)) + 1 ^ succ b * b ^ (succ b - succ b) * ↑(choose (succ b) (succ b))) : 1 + (b * succ b + b ^ succ b) ≤ (1 + b) ^ succ b := by rw [add_pow 1 b b.succ] rw [Finset.sum_range_succ _ b.succ] exact h₁ lemma imo_2022_p5_1_4 (b : ℕ) -- (p : ℕ) (h₀ : 0 < b) : -- (hbp : b < p) : b * succ b + b ^ succ b ≤ Finset.sum (Finset.range (succ b)) fun x => b ^ (succ b - x) * choose (succ b) x := by rw [Finset.sum_range_succ _ b] rw [succ_eq_add_one] simp rw [add_comm _ (b * (b + 1))] simp have gb: b = b - 1 + 1 := by exact (Nat.sub_eq_iff_eq_add h₀).mp rfl nth_rewrite 3 [gb] rw [Finset.sum_range_succ' _ (b-1)] simp lemma imo_2022_p5_1_5 (b : ℕ) -- (p : ℕ) -- (h₀ : 0 < b) -- (hbp : b < p) (h₁ : b * (b + 1) + b ^ (b + 1) ≤ (Finset.sum (Finset.range b) fun x => b ^ (b + 1 - x) * choose (b + 1) x) + b ^ (b + 1 - b) * choose (b + 1) b) : 1 + (b * succ b + b ^ succ b) ≤ (1 + b) ^ succ b := by rw [add_pow 1 b b.succ] rw [Finset.sum_range_succ _ b.succ] simp rw [add_comm (∑ x ∈ Finset.range (b + 1), b ^ (b + 1 - x) * (b + 1).choose x) 1] simp rw [Finset.sum_range_succ _ b] exact h₁ lemma imo_2022_p5_1_6 (b : ℕ) -- (p : ℕ) -- (h₀ : 0 < b) -- (hbp : b < p) (h₁ : b * succ b + b ^ succ b ≤ Finset.sum (Finset.range (succ b)) fun x => b ^ (succ b - x) * choose (succ b) x) : 1 + (b * succ b + b ^ succ b) ≤ (Finset.sum (Finset.range (succ b)) fun x => 1 ^ x * b ^ (succ b - x) * ↑(choose (succ b) x)) + 1 ^ succ b * b ^ (succ b - succ b) * ↑(choose (succ b) (succ b)) := by simp rw [add_comm (∑ x ∈ Finset.range (b + 1), b ^ (b + 1 - x) * (b + 1).choose x) 1] simp exact h₁ lemma imo_2022_p5_1_7 (b : ℕ) -- (p : ℕ) (h₀ : 0 < b) : -- (hbp : b < p) : b * succ b + b ^ succ b ≤ (Finset.sum (Finset.range b) fun x => b ^ (succ b - x) * choose (succ b) x) + b ^ (succ b - b) * choose (succ b) b := by rw [succ_eq_add_one] simp rw [add_comm _ (b * (b + 1))] simp have gb: b = b - 1 + 1 := by exact (Nat.sub_eq_iff_eq_add h₀).mp rfl nth_rewrite 3 [gb] rw [Finset.sum_range_succ' _ (b-1)] simp lemma imo_2022_p5_1_8 (b : ℕ) -- (p : ℕ) (h₀ : 0 < b) : -- (hbp : b < p) : b * (b + 1) + b ^ (b + 1) ≤ (Finset.sum (Finset.range b) fun x => b ^ (b + 1 - x) * choose (b + 1) x) + b * (b + 1) := by rw [add_comm _ (b * (b + 1))] simp have gb: b = b - 1 + 1 := by exact (Nat.sub_eq_iff_eq_add h₀).mp rfl nth_rewrite 3 [gb] rw [Finset.sum_range_succ' _ (b-1)] simp lemma imo_2022_p5_1_9 (b : ℕ) -- (p : ℕ) (h₀ : 0 < b) : -- (hbp : b < p) : b ^ (b + 1) ≤ Finset.sum (Finset.range b) fun x => b ^ (b + 1 - x) * choose (b + 1) x := by have gb: b = b - 1 + 1 := by exact (Nat.sub_eq_iff_eq_add h₀).mp rfl nth_rewrite 3 [gb] rw [Finset.sum_range_succ' _ (b-1)] simp lemma imo_2022_p5_1_10 (b : ℕ) -- (p : ℕ) -- (h₀ : 0 < b) -- (hbp : b < p) (gb : b = b - 1 + 1) : b ^ (b + 1) ≤ Finset.sum (Finset.range b) fun x => b ^ (b + 1 - x) * choose (b + 1) x := by nth_rewrite 3 [gb] rw [Finset.sum_range_succ' _ (b-1)] simp lemma imo_2022_p5_1_11 (b : ℕ) : -- (p : ℕ) -- (h₀ : 0 < b) -- (hbp : b < p) -- (gb : b = b - 1 + 1) : b ^ (b + 1) ≤ Finset.sum (Finset.range (b - 1 + 1)) fun x => b ^ (b + 1 - x) * choose (b + 1) x := by rw [Finset.sum_range_succ' _ (b-1)] simp lemma imo_2022_p5_1_12 (b : ℕ) : -- (p : ℕ) -- (h₀ : 0 < b) -- (hbp : b < p) : ∀ (n : ℕ), succ b ≤ n → 1 + (b * n + b ^ n) ≤ (1 + b) ^ n → 1 + (b * (n + 1) + b ^ (n + 1)) ≤ (1 + b) ^ (n + 1) := by intros n _ h₂ nth_rewrite 2 [pow_add] rw [pow_one] have h₃: (1 + (b * n + b ^ n)) * (1 + b) ≤ ((1 + b) ^ n) * (1 + b) := by exact mul_le_mul_right' h₂ (1 + b) have h₄: 1 + (b * (n + 1) + b ^ (n + 1)) ≤ (1 + (b * n + b ^ n)) * (1 + b) := by ring_nf rw [Nat.add_assoc _ (b ^ 2 * n) (b ^ n)] exact Nat.le_add_right (1 + b + b * b ^ n + b * n) (b ^ 2 * n + b ^ n) exact le_trans h₄ h₃ lemma imo_2022_p5_1_13 (b : ℕ) -- (p : ℕ) -- (h₀ : 0 < b) -- (hbp : b < p) (n : ℕ) -- (hmn : succ b ≤ n) (h₂ : 1 + (b * n + b ^ n) ≤ (1 + b) ^ n) : 1 + (b * (n + 1) + b ^ (n + 1)) ≤ (1 + b) ^ n * (1 + b) := by have h₃: (1 + (b * n + b ^ n)) * (1 + b) ≤ ((1 + b) ^ n) * (1 + b) := by exact mul_le_mul_right' h₂ (1 + b) have h₄: 1 + (b * (n + 1) + b ^ (n + 1)) ≤ (1 + (b * n + b ^ n)) * (1 + b) := by ring_nf rw [Nat.add_assoc _ (b ^ 2 * n) (b ^ n)] exact Nat.le_add_right (1 + b + b * b ^ n + b * n) (b ^ 2 * n + b ^ n) exact le_trans h₄ h₃ lemma imo_2022_p5_1_14 (b : ℕ) -- (p : ℕ) -- (h₀ : 0 < b) -- (hbp : b < p) (n : ℕ) : -- (h₂ : 1 + (b * n + b ^ n) ≤ (1 + b) ^ n) -- (h₃ : (1 + (b * n + b ^ n)) * (1 + b) ≤ ((1 + b) ^ n) * (1 + b)) : 1 + (b * (n + 1) + b ^ (n + 1)) ≤ (1 + (b * n + b ^ n)) * (1 + b) := by have h₄: 1 + (b * (n + 1) + b ^ (n + 1)) ≤ (1 + (b * n + b ^ n)) * (1 + b) := by ring_nf rw [Nat.add_assoc _ (b ^ 2 * n) (b ^ n)] exact Nat.le_add_right (1 + b + b * b ^ n + b * n) (b ^ 2 * n + b ^ n) refine le_trans h₄ ?_ linarith lemma imo_2022_p5_1_15 (b : ℕ) -- (p : ℕ) -- (h₀ : 0 < b) -- (hbp : b < p) (n : ℕ) : -- (hmn : succ b ≤ n) -- (h₂ : 1 + (b * n + b ^ n) ≤ (1 + b) ^ n) -- (h₃ : (1 + (b * n + b ^ n)) * (1 + b) ≤ (1 + b) ^ n * (1 + b)) : 1 + b + b * b ^ n + b * n ≤ 1 + b + b * b ^ n + b * n + b ^ 2 * n + b ^ n := by ring_nf rw [Nat.add_assoc _ (b ^ 2 * n) (b ^ n)] exact Nat.le_add_right (1 + b + b * b ^ n + b * n) (b ^ 2 * n + b ^ n) lemma imo_2022_p5_2 (n : ℕ) (hi : n ! ≤ n ^ n) : (succ n)! ≤ succ n ^ succ n := by by_cases hnp: 0 < n . rw [ factorial_succ, succ_eq_add_one, pow_add, pow_one, mul_comm ] refine mul_le_mul_right (n + 1) ?_ -- have h₁: n.factorial ≤ n ^ n, -- { exact hi hnp }, have h₂: n^ n ≤ (n + 1)^n := by refine (Nat.pow_le_pow_iff_left ?_).mpr ?_ . linarith . linarith exact le_trans hi h₂ . push_neg at hnp interval_cases n simp lemma imo_2022_p5_2_1 (n : ℕ) (hi : n ! ≤ n ^ n) (hnp : 0 < n) : (succ n)! ≤ succ n ^ succ n := by rw [ factorial_succ, succ_eq_add_one, pow_add, pow_one, mul_comm ] refine mul_le_mul_right (n + 1) ?_ have h₂: n^ n ≤ (n + 1)^n := by refine (Nat.pow_le_pow_iff_left ?_).mpr ?_ . linarith . linarith exact le_trans hi h₂ lemma imo_2022_p5_2_2 (n : ℕ) (hi : n ! ≤ n ^ n) (hnp : ¬0 < n) : (succ n)! ≤ succ n ^ succ n := by push_neg at hnp interval_cases n simp lemma imo_2022_p5_2_3 (n : ℕ) (hi : n ! ≤ n ^ n) (hnp : 0 < n) : -- (h₁: (succ n)! ≤ succ n ^ succ n) : n ! * (n + 1) ≤ (n + 1) ^ n * (n + 1) := by refine mul_le_mul_right (n + 1) ?_ have h₂: n^ n ≤ (n + 1)^n := by refine (Nat.pow_le_pow_iff_left ?_).mpr ?_ . linarith . linarith exact le_trans hi h₂ lemma imo_2022_p5_2_4 (n : ℕ) (hi : n ! ≤ n ^ n) (hnp : 0 < n) : n ! ≤ (n + 1) ^ n := by have h₂: n^ n ≤ (n + 1)^n := by refine (Nat.pow_le_pow_iff_left ?_).mpr ?_ . linarith . linarith exact le_trans hi h₂ lemma imo_2022_p5_2_5 (n : ℕ) -- (hi : n ! ≤ n ^ n) (hnp : 0 < n) : n ^ n ≤ (n + 1) ^ n := by refine (Nat.pow_le_pow_iff_left ?_).mpr ?_ . linarith . linarith lemma imo_2022_p5_3 (a b p: ℕ) -- (h₀: 0 < a ∧ 0 < b) (hp: Nat.Prime p) (h₁: a ^ p = b.factorial + p) (hbp: p ≤ b) : (p ∣ a) := by have h₂: p ∣ b.factorial := by exact Nat.dvd_factorial (Nat.Prime.pos hp) hbp have h₃: p ∣ b.factorial + p := by exact Nat.dvd_add_self_right.mpr h₂ have h₄: p ∣ a^p := by rw [h₁] exact h₃ exact Nat.Prime.dvd_of_dvd_pow hp h₄ lemma imo_2022_p5_3_1 (a b p : ℕ) (hp : Nat.Prime p) (h₁ : a ^ p = b ! + p) -- (hbp : p ≤ b) (h₂ : p ∣ b !) : p ∣ a := by have h₃: p ∣ b.factorial + p := by exact Nat.dvd_add_self_right.mpr h₂ have h₄: p ∣ a^p := by rw [h₁] exact h₃ exact Nat.Prime.dvd_of_dvd_pow hp h₄ lemma imo_2022_p5_3_2 (a b p : ℕ) (hp : Nat.Prime p) (h₁ : a ^ p = b ! + p) -- (hbp : p ≤ b) -- (h₂ : p ∣ b !) (h₃ : p ∣ b ! + p) : p ∣ a := by have h₄: p ∣ a^p := by rw [h₁] exact h₃ exact Nat.Prime.dvd_of_dvd_pow hp h₄ lemma imo_2022_p5_4 (a b : ℕ) (h₀: 2 ≤ a) (h₁: a < b) : (a + b < a * b ) := by have h₂: a + b < b + b := by exact add_lt_add_right h₁ b have h₃: b + b ≤ a * b := by rw [← two_mul] exact mul_le_mul_right' h₀ b exact gt_of_ge_of_gt h₃ h₂ lemma imo_2022_p5_4_1 (a b : ℕ) (h₀ : 2 ≤ a) -- (h₁ : a < b) (h₂ : a + b < b + b) : a + b < a * b := by have h₃: b + b ≤ a * b := by rw [← two_mul] exact mul_le_mul_right' h₀ b exact gt_of_ge_of_gt h₃ h₂ lemma imo_2022_p5_4_2 (a b : ℕ) (h₀ : 2 ≤ a) : -- (h₁ : a < b) -- (h₂ : a + b < b + b) : b + b ≤ a * b := by rw [← two_mul] exact mul_le_mul_right' h₀ b lemma imo_2022_p5_5 (p: ℕ) : (Finset.Ico p (2 * p - 1)).prod (fun x => x + 1) = (Finset.range (p - 1)).prod (fun x => p + (x + 1)) := by rw [Finset.prod_Ico_eq_prod_range _ (p) (2 * p - 1)] have h₀: 2 * p - 1 - p = p - 1 := by omega rw [h₀] exact rfl lemma imo_2022_p5_5_1 (p : ℕ) : (Finset.prod (Finset.range (2 * p - 1 - p)) fun k => p + k + 1) = Finset.prod (Finset.range (p - 1)) fun x => p + (x + 1) := by have h₀: 2 * p - 1 - p = p - 1 := by omega rw [h₀] exact rfl lemma imo_2022_p5_5_2 (p : ℕ) (h₀ : 2 * p - 1 - p = p - 1) : (Finset.prod (Finset.range (2 * p - 1 - p)) fun k => p + k + 1) = Finset.prod (Finset.range (p - 1)) fun x => p + (x + 1) := by rw [h₀] exact rfl lemma imo_2022_p5_6 (p: ℕ) (hp: 2 ≤ p) : (Finset.range (p - 1)).prod (fun (x : ℕ) => x + 1) = (Finset.range (p - 1)).prod (fun (x : ℕ) => p - (x + 1)) := by refine Nat.le_induction ?_ ?_ p hp . norm_num . intros n hn2 h₀ simp at * have hn: 0 < n := by exact lt_of_succ_lt hn2 rw [← Nat.mul_factorial_pred hn, h₀] let f: (ℕ → ℕ) := fun (x : ℕ) => n - x have h₁: (Finset.range n).prod f = (Finset.range 1).prod f * (Finset.Ico 1 n).prod f := by exact (Finset.prod_range_mul_prod_Ico (fun k => n - k) hn).symm rw [h₁] have h₂: (Finset.range 1).prod f = n := by exact Finset.prod_range_one fun k => n - k rw [h₂] simp left rw [Finset.prod_Ico_eq_prod_range f 1 n] ring_nf exact rfl lemma imo_2022_p5_6_1 : -- (p : ℕ) -- (hp : 2 ≤ p) : ∀ (n : ℕ), 2 ≤ n → ((Finset.prod (Finset.range (n - 1)) fun x => x + 1) = Finset.prod (Finset.range (n - 1)) fun x => n - (x + 1)) → (Finset.prod (Finset.range (n + 1 - 1)) fun x => x + 1) = Finset.prod (Finset.range (n + 1 - 1)) fun x => n + 1 - (x + 1) := by intros n hn2 h₀ simp at * have hn: 0 < n := by exact lt_of_succ_lt hn2 rw [← Nat.mul_factorial_pred hn, h₀] let f: (ℕ → ℕ) := fun (x : ℕ) => n - x have h₁: (Finset.range n).prod f = (Finset.range 1).prod f * (Finset.Ico 1 n).prod f := by exact (Finset.prod_range_mul_prod_Ico (fun k => n - k) hn).symm rw [h₁] have h₂: (Finset.range 1).prod f = n := by exact Finset.prod_range_one fun k => n - k rw [h₂] simp left rw [Finset.prod_Ico_eq_prod_range f 1 n] ring_nf exact rfl lemma imo_2022_p5_6_2 -- (p : ℕ) -- (hp : 2 ≤ p) (n : ℕ) (hn2 : 2 ≤ n) (h₀ : (n - 1)! = Finset.prod (Finset.range (n - 1)) fun x => n - (x + 1)) : n ! = Finset.prod (Finset.range n) fun x => n - x := by have hn: 0 < n := by exact lt_of_succ_lt hn2 rw [← Nat.mul_factorial_pred hn, h₀] let f: (ℕ → ℕ) := fun (x : ℕ) => n - x have h₁: (Finset.range n).prod f = (Finset.range 1).prod f * (Finset.Ico 1 n).prod f := by exact (Finset.prod_range_mul_prod_Ico (fun k => n - k) hn).symm rw [h₁] have h₂: (Finset.range 1).prod f = n := by exact Finset.prod_range_one fun k => n - k rw [h₂] simp left rw [Finset.prod_Ico_eq_prod_range f 1 n] ring_nf exact rfl lemma imo_2022_p5_6_3 -- (p : ℕ) -- (hp : 2 ≤ p) (n : ℕ) -- (hn2 : 2 ≤ n) -- (h₀ : (n - 1)! = Finset.prod (Finset.range (n - 1)) fun x => n - (x + 1)) (hn : 0 < n) : n * (Finset.prod (Finset.range (n - 1)) fun x => n - (x + 1)) = Finset.prod (Finset.range n) fun x => n - x := by let f: (ℕ → ℕ) := fun (x : ℕ) => n - x have h₁: (Finset.range n).prod f = (Finset.range 1).prod f * (Finset.Ico 1 n).prod f := by exact (Finset.prod_range_mul_prod_Ico (fun k => n - k) hn).symm rw [h₁] have h₂: (Finset.range 1).prod f = n := by exact Finset.prod_range_one fun k => n - k rw [h₂] simp left rw [Finset.prod_Ico_eq_prod_range f 1 n] ring_nf exact rfl lemma imo_2022_p5_6_4 -- (p : ℕ) -- (hp : 2 ≤ p) (n : ℕ) -- (hn2 : 2 ≤ n) -- (h₀ : (n - 1)! = Finset.prod (Finset.range (n - 1)) fun x => n - (x + 1)) (hn : 0 < n) : -- (f : ℕ → ℕ) : -- (hf: f = fun x => n - x) : n * (Finset.prod (Finset.range (n - 1)) fun x => n - (x + 1)) = Finset.prod (Finset.range n) fun x => n - x := by have h₁: (Finset.range n).prod (fun x => n - x) = (Finset.range 1).prod (fun x => n - x) * (Finset.Ico 1 n).prod (fun x => n - x) := by exact (Finset.prod_range_mul_prod_Ico (fun k => n - k) hn).symm rw [h₁] have h₂: (Finset.range 1).prod (fun x => n - x) = n := by -- rw [hf] exact Finset.prod_range_one fun k => n - k rw [h₂] simp left rw [Finset.prod_Ico_eq_prod_range (fun x => n - x) 1 n] ring_nf lemma imo_2022_p5_6_5 -- (p : ℕ) -- (hp : 2 ≤ p) (n : ℕ) -- (hn2 : 2 ≤ n) -- (h₀ : (n - 1)! = Finset.prod (Finset.range (n - 1)) fun x => n - (x + 1)) -- (hn : 0 < n) (f : ℕ → ℕ) (hf: f = fun x => n - x) (h₁ : Finset.prod (Finset.range n) f = Finset.prod (Finset.range 1) f * Finset.prod (Finset.Ico 1 n) f) : n * (Finset.prod (Finset.range (n - 1)) fun x => n - (x + 1)) = Finset.prod (Finset.range n) fun x => n - x := by rw [← hf, h₁] have h₂: (Finset.range 1).prod f = n := by rw [hf] exact Finset.prod_range_one fun k => n - k rw [h₂] simp left rw [Finset.prod_Ico_eq_prod_range f 1 n] ring_nf rw [hf] lemma imo_2022_p5_6_6 -- (p : ℕ) -- (hp : 2 ≤ p) (n : ℕ) -- (hn2 : 2 ≤ n) -- (h₀ : (n - 1)! = Finset.prod (Finset.range (n - 1)) fun x => n - (x + 1)) -- (hn : 0 < n) (f : ℕ → ℕ) (hf: f = fun x => n - x) -- (h₁ : Finset.prod (Finset.range n) f = Finset.prod (Finset.range 1) f * Finset.prod (Finset.Ico 1 n) f) (h₂ : Finset.prod (Finset.range 1) f = n) : n * (Finset.prod (Finset.range (n - 1)) fun x => n - (x + 1)) = Finset.prod (Finset.range 1) f * Finset.prod (Finset.Ico 1 n) f := by rw [h₂] simp left rw [Finset.prod_Ico_eq_prod_range f 1 n] ring_nf rw [hf] lemma imo_2022_p5_6_7 -- (p : ℕ) -- (hp : 2 ≤ p) (n : ℕ) -- (hn2 : 2 ≤ n) -- (h₀ : (n - 1)! = Finset.prod (Finset.range (n - 1)) fun x => n - (x + 1)) -- (hn : 0 < n) (f : ℕ → ℕ) (hf: f = fun x => n - x) : -- (h₁ : Finset.prod (Finset.range n) f = Finset.prod (Finset.range 1) f * Finset.prod (Finset.Ico 1 n) f) -- (h₂ : Finset.prod (Finset.range 1) f = n) : (Finset.prod (Finset.range (n - 1)) fun x => n - (x + 1)) = Finset.prod (Finset.Ico 1 n) f ∨ n = 0 := by left rw [Finset.prod_Ico_eq_prod_range f 1 n] ring_nf rw [hf] lemma imo_2022_p5_7 (b p: ℕ) -- (h₀: 0 < b) (hp: Nat.Prime p) (hb2p: b < 2 * p) : b.factorial + p < p ^ (2 * p) := by have h₁: b.factorial ≤ (2*p - 1).factorial := by refine factorial_le ?_ exact le_pred_of_lt hb2p have gp: 2 ≤ p := by exact Prime.two_le hp have gp1: (p - 1) + 1 = p := by refine Nat.sub_add_cancel ?_ exact one_le_of_lt gp let f: (ℕ → ℕ) := (fun (x : ℕ) => x + 1) have h₂: (Finset.range (2 * p - 1)).prod f = (Finset.range (p - 1)).prod (fun (x : ℕ) => p^2 - (x+1)^2) * p := by -- break the prod into three segments rang(p-1) + p + (p+1) until 2p-1 have g₀: (Finset.range (2 * p - 1)).prod f = (Finset.range ((p - 1) + 1)).prod f * (Finset.Ico ((p - 1) + 1) (2 * p - 1)).prod f := by symm refine Finset.prod_range_mul_prod_Ico f ?_ rw [gp1] have gg₀: p + 2 - 1 ≤ 2 * p - 1 := by refine Nat.sub_le_sub_right ?_ 1 rw [add_comm] exact add_le_mul (by norm_num) gp exact le_of_lt gg₀ have g₁: (Finset.range ((p - 1) + 1)).prod (fun (x : ℕ) => x + 1) = (Finset.range (p - 1)).prod (fun (x : ℕ) => x + 1) * ((p - 1) + 1) := by exact Finset.prod_range_succ _ (p - 1) rw [g₁] at g₀ nth_rewrite 2 [mul_comm] at g₀ rw [← mul_assoc] at g₀ rw [gp1] at g₀ g₁ have g₂: (Finset.Ico ((p - 1) + 1) (2 * p - 1)).prod (fun (x : ℕ) => x + 1) = (Finset.range (p - 1)).prod (fun (x : ℕ) => p + (x+1)) := by rw [gp1] exact imo_2022_p5_5 p have g₃: (Finset.range (p - 1)).prod (fun (x : ℕ) => x + 1) = (Finset.range (p - 1)).prod (fun (x : ℕ) => p - (x+1)) := by exact imo_2022_p5_6 p gp rw [gp1] at g₂ rw [g₂,g₃] at g₀ have g₄: (Finset.range (p - 1)).prod (fun (x : ℕ) => p + (x+1)) * (Finset.range (p - 1)).prod (fun (x : ℕ) => p - (x+1)) = (Finset.range (p - 1)).prod (fun (x : ℕ) => (p + (x+1)) * (p - (x+1))) := by symm refine Finset.prod_mul_distrib have g₅: (fun (x : ℕ) => p ^ 2 - (x+1) ^ 2) = (fun (x : ℕ) => (p + (x+1)) * (p - (x+1))) := by ext1 x exact Nat.sq_sub_sq p (x + 1) rw [g₄,← g₅] at g₀ exact g₀ have h₃: (Finset.range (p - 1)).prod (fun (x : ℕ) => p^2 - (x+1)^2) * p ≤ (p^2)^(Finset.range (p - 1)).card * p := by refine Nat.mul_le_mul_right ?_ ?_ refine Finset.prod_le_pow_card (Finset.range (p - 1)) ?_ (p^2) ?_ intros x _ exact (p ^ 2).sub_le ((x + 1) ^ 2) simp at * have h₄: b.factorial + p ≤ (p ^ 2) ^ (p - 1) * p + p := by refine add_le_add_right ?_ p refine le_trans ?_ h₃ rw [← h₂] rw [Finset.prod_range_add_one_eq_factorial] exact h₁ have h₅: b.factorial + p < (p ^ 2) ^ (p - 1) * p * p := by refine lt_of_le_of_lt h₄ ?_ rw [add_comm] nth_rewrite 2 [mul_comm] refine imo_2022_p5_4 p ((p ^ 2) ^ (p - 1) * p) gp ?_ refine lt_mul_left (by linarith) ?_ rw [← pow_mul] refine Nat.one_lt_pow ?_ (Nat.lt_of_succ_le gp) refine Nat.mul_ne_zero (by norm_num) ?_ exact Nat.sub_ne_zero_iff_lt.mpr gp rw [mul_assoc _ p p, ← pow_two p] at h₅ rw [← Nat.pow_succ, succ_eq_add_one, gp1] at h₅ rw [Nat.pow_mul] exact h₅ lemma imo_2022_p5_7_1 (b p : ℕ) -- (h₀ : 0 < b) -- (hp : Nat.Prime p) (hb2p : b < 2 * p) : b ! ≤ (2 * p - 1)! := by refine factorial_le ?_ exact le_pred_of_lt hb2p lemma imo_2022_p5_7_2 (b p : ℕ) -- (h₀ : 0 < b) -- (hp : Nat.Prime p) -- (hb2p : b < 2 * p) (h₁ : b ! ≤ (2 * p - 1)!) (gp : 2 ≤ p) : b ! + p < p ^ (2 * p) := by have gp1: (p - 1) + 1 = p := by refine Nat.sub_add_cancel ?_ exact one_le_of_lt gp let f: (ℕ → ℕ) := (fun (x : ℕ) => x + 1) have h₂: (Finset.range (2 * p - 1)).prod f = (Finset.range (p - 1)).prod (fun (x : ℕ) => p^2 - (x+1)^2) * p := by -- break the prod into three segments rang(p-1) + p + (p+1) until 2p-1 have g₀: (Finset.range (2 * p - 1)).prod f = (Finset.range ((p - 1) + 1)).prod f * (Finset.Ico ((p - 1) + 1) (2 * p - 1)).prod f := by symm refine Finset.prod_range_mul_prod_Ico f ?_ rw [gp1] have gg₀: p + 2 - 1 ≤ 2 * p - 1 := by refine Nat.sub_le_sub_right ?_ 1 rw [add_comm] exact add_le_mul (by norm_num) gp exact le_of_lt gg₀ have g₁: (Finset.range ((p - 1) + 1)).prod (fun (x : ℕ) => x + 1) = (Finset.range (p - 1)).prod (fun (x : ℕ) => x + 1) * ((p - 1) + 1) := by exact Finset.prod_range_succ _ (p - 1) rw [g₁] at g₀ nth_rewrite 2 [mul_comm] at g₀ rw [← mul_assoc] at g₀ rw [gp1] at g₀ g₁ have g₂: (Finset.Ico ((p - 1) + 1) (2 * p - 1)).prod (fun (x : ℕ) => x + 1) = (Finset.range (p - 1)).prod (fun (x : ℕ) => p + (x+1)) := by rw [gp1] exact imo_2022_p5_5 p have g₃: (Finset.range (p - 1)).prod (fun (x : ℕ) => x + 1) = (Finset.range (p - 1)).prod (fun (x : ℕ) => p - (x+1)) := by exact imo_2022_p5_6 p gp rw [gp1] at g₂ rw [g₂,g₃] at g₀ have g₄: (Finset.range (p - 1)).prod (fun (x : ℕ) => p + (x+1)) * (Finset.range (p - 1)).prod (fun (x : ℕ) => p - (x+1)) = (Finset.range (p - 1)).prod (fun (x : ℕ) => (p + (x+1)) * (p - (x+1))) := by symm refine Finset.prod_mul_distrib have g₅: (fun (x : ℕ) => p ^ 2 - (x+1) ^ 2) = (fun (x : ℕ) => (p + (x+1)) * (p - (x+1))) := by ext1 x exact Nat.sq_sub_sq p (x + 1) rw [g₄,← g₅] at g₀ exact g₀ have h₃: (Finset.range (p - 1)).prod (fun (x : ℕ) => p^2 - (x+1)^2) * p ≤ (p^2)^(Finset.range (p - 1)).card * p := by refine Nat.mul_le_mul_right ?_ ?_ refine Finset.prod_le_pow_card (Finset.range (p - 1)) ?_ (p^2) ?_ intros x _ exact (p ^ 2).sub_le ((x + 1) ^ 2) simp at * have h₄: b.factorial + p ≤ (p ^ 2) ^ (p - 1) * p + p := by refine add_le_add_right ?_ p refine le_trans ?_ h₃ rw [← h₂] rw [Finset.prod_range_add_one_eq_factorial] exact h₁ have h₅: b.factorial + p < (p ^ 2) ^ (p - 1) * p * p := by refine lt_of_le_of_lt h₄ ?_ rw [add_comm] nth_rewrite 2 [mul_comm] refine imo_2022_p5_4 p ((p ^ 2) ^ (p - 1) * p) gp ?_ refine lt_mul_left (by linarith) ?_ rw [← pow_mul] refine Nat.one_lt_pow ?_ (Nat.lt_of_succ_le gp) refine Nat.mul_ne_zero (by norm_num) ?_ exact Nat.sub_ne_zero_iff_lt.mpr gp rw [mul_assoc _ p p, ← pow_two p] at h₅ rw [← Nat.pow_succ, succ_eq_add_one, gp1] at h₅ rw [Nat.pow_mul] exact h₅ lemma imo_2022_p5_7_3 (b p : ℕ) -- (h₀ : 0 < b) -- (hp : Nat.Prime p) -- (hb2p : b < 2 * p) (h₁ : b ! ≤ (2 * p - 1)!) (gp : 2 ≤ p) (gp1 : p - 1 + 1 = p) (f : ℕ → ℕ) (hf : f = fun x => x + 1) : b ! + p < p ^ (2 * p) := by have h₂: (Finset.range (2 * p - 1)).prod f = (Finset.range (p - 1)).prod (fun (x : ℕ) => p^2 - (x+1)^2) * p := by -- important -- break the prod into three segments rang(p-1) + p + (p+1) until 2p-1 have g₀: (Finset.range (2 * p - 1)).prod f = (Finset.range ((p - 1) + 1)).prod f * (Finset.Ico ((p - 1) + 1) (2 * p - 1)).prod f := by symm refine Finset.prod_range_mul_prod_Ico f ?_ rw [gp1] have gg₀: p + 2 - 1 ≤ 2 * p - 1 := by refine Nat.sub_le_sub_right ?_ 1 rw [add_comm] exact add_le_mul (by norm_num) gp exact le_of_lt gg₀ have g₁: (Finset.range ((p - 1) + 1)).prod (fun (x : ℕ) => x + 1) = (Finset.range (p - 1)).prod (fun (x : ℕ) => x + 1) * ((p - 1) + 1) := by exact Finset.prod_range_succ _ (p - 1) rw [hf, g₁] at g₀ nth_rewrite 2 [mul_comm] at g₀ rw [← mul_assoc] at g₀ rw [gp1] at g₀ g₁ have g₂: (Finset.Ico ((p - 1) + 1) (2 * p - 1)).prod (fun (x : ℕ) => x + 1) = (Finset.range (p - 1)).prod (fun (x : ℕ) => p + (x+1)) := by rw [gp1] exact imo_2022_p5_5 p have g₃: (Finset.range (p - 1)).prod (fun (x : ℕ) => x + 1) = (Finset.range (p - 1)).prod (fun (x : ℕ) => p - (x+1)) := by exact imo_2022_p5_6 p gp rw [gp1] at g₂ rw [g₂,g₃] at g₀ have g₄: (Finset.range (p - 1)).prod (fun (x : ℕ) => p + (x+1)) * (Finset.range (p - 1)).prod (fun (x : ℕ) => p - (x+1)) = (Finset.range (p - 1)).prod (fun (x : ℕ) => (p + (x+1)) * (p - (x+1))) := by symm refine Finset.prod_mul_distrib have g₅: (fun (x : ℕ) => p ^ 2 - (x+1) ^ 2) = (fun (x : ℕ) => (p + (x+1)) * (p - (x+1))) := by ext1 x exact Nat.sq_sub_sq p (x + 1) rw [g₄,← g₅, ← hf] at g₀ exact g₀ have h₃: (Finset.range (p - 1)).prod (fun (x : ℕ) => p^2 - (x+1)^2) * p ≤ (p^2)^(Finset.range (p - 1)).card * p := by refine Nat.mul_le_mul_right ?_ ?_ refine Finset.prod_le_pow_card (Finset.range (p - 1)) ?_ (p^2) ?_ intros x _ exact (p ^ 2).sub_le ((x + 1) ^ 2) simp at * have h₄: b.factorial + p ≤ (p ^ 2) ^ (p - 1) * p + p := by refine add_le_add_right ?_ p refine le_trans ?_ h₃ rw [← h₂, hf] rw [Finset.prod_range_add_one_eq_factorial] exact h₁ have h₅: b.factorial + p < (p ^ 2) ^ (p - 1) * p * p := by refine lt_of_le_of_lt h₄ ?_ rw [add_comm] nth_rewrite 2 [mul_comm] refine imo_2022_p5_4 p ((p ^ 2) ^ (p - 1) * p) gp ?_ refine lt_mul_left (by linarith) ?_ rw [← pow_mul] refine Nat.one_lt_pow ?_ (Nat.lt_of_succ_le gp) refine Nat.mul_ne_zero (by norm_num) ?_ exact Nat.sub_ne_zero_iff_lt.mpr gp rw [mul_assoc _ p p, ← pow_two p] at h₅ rw [← Nat.pow_succ, succ_eq_add_one, gp1] at h₅ rw [Nat.pow_mul] exact h₅ lemma imo_2022_p5_7_4 -- (b : ℕ) (p : ℕ) -- (h₀ : 0 < b) -- (hp : Nat.Prime p) -- (hb2p : b < 2 * p) -- (h₁ : b ! ≤ (2 * p - 1)!) (gp : 2 ≤ p) (gp1 : p - 1 + 1 = p) (f : ℕ → ℕ) (hf : f = fun x => x + 1) : Finset.prod (Finset.range (2 * p - 1)) f = (Finset.prod (Finset.range (p - 1)) fun x => p ^ 2 - (x + 1) ^ 2) * p := by have g₀: (Finset.range (2 * p - 1)).prod f = (Finset.range ((p - 1) + 1)).prod f * (Finset.Ico ((p - 1) + 1) (2 * p - 1)).prod f := by symm refine Finset.prod_range_mul_prod_Ico f ?_ rw [gp1] have gg₀: p + 2 - 1 ≤ 2 * p - 1 := by refine Nat.sub_le_sub_right ?_ 1 rw [add_comm] exact add_le_mul (by norm_num) gp exact le_of_lt gg₀ have g₁: (Finset.range ((p - 1) + 1)).prod (fun (x : ℕ) => x + 1) = (Finset.range (p - 1)).prod (fun (x : ℕ) => x + 1) * ((p - 1) + 1) := by exact Finset.prod_range_succ _ (p - 1) rw [hf, g₁] at g₀ nth_rewrite 2 [mul_comm] at g₀ rw [← mul_assoc] at g₀ rw [gp1] at g₀ g₁ have g₂: (Finset.Ico ((p - 1) + 1) (2 * p - 1)).prod (fun (x : ℕ) => x + 1) = (Finset.range (p - 1)).prod (fun (x : ℕ) => p + (x+1)) := by rw [gp1] exact imo_2022_p5_5 p have g₃: (Finset.range (p - 1)).prod (fun (x : ℕ) => x + 1) = (Finset.range (p - 1)).prod (fun (x : ℕ) => p - (x+1)) := by exact imo_2022_p5_6 p gp rw [gp1] at g₂ rw [g₂,g₃] at g₀ have g₄: (Finset.range (p - 1)).prod (fun (x : ℕ) => p + (x+1)) * (Finset.range (p - 1)).prod (fun (x : ℕ) => p - (x+1)) = (Finset.range (p - 1)).prod (fun (x : ℕ) => (p + (x+1)) * (p - (x+1))) := by symm refine Finset.prod_mul_distrib have g₅: (fun (x : ℕ) => p ^ 2 - (x+1) ^ 2) = (fun (x : ℕ) => (p + (x+1)) * (p - (x+1))) := by ext1 x exact Nat.sq_sub_sq p (x + 1) rw [g₄,← g₅, ← hf] at g₀ exact g₀ lemma imo_2022_p5_7_5 -- (b : ℕ) (p : ℕ) -- (h₀ : 0 < b) -- (hp : Nat.Prime p) -- (hb2p : b < 2 * p) -- (h₁ : b ! ≤ (2 * p - 1)!) (gp : 2 ≤ p) (gp1 : p - 1 + 1 = p) (f : ℕ → ℕ) : -- (hf : f = fun x => x + 1) : Finset.prod (Finset.range (2 * p - 1)) f = Finset.prod (Finset.range (p - 1 + 1)) f * Finset.prod (Finset.Ico (p - 1 + 1) (2 * p - 1)) f := by symm refine Finset.prod_range_mul_prod_Ico f ?_ rw [gp1] have gg₀: p + 2 - 1 ≤ 2 * p - 1 := by refine Nat.sub_le_sub_right ?_ 1 rw [add_comm] exact add_le_mul (by norm_num) gp exact le_of_lt gg₀ lemma imo_2022_p5_7_6 -- (b : ℕ) (p : ℕ) -- (h₀ : 0 < b) -- (hp : Nat.Prime p) -- (hb2p : b < 2 * p) -- (h₁ : b ! ≤ (2 * p - 1)!) (gp : 2 ≤ p) (gp1 : p - 1 + 1 = p) : -- (f : ℕ → ℕ) -- (hf : f = fun x => x + 1) : p - 1 + 1 ≤ 2 * p - 1 := by have h₂: p - 1 + 1 < p + 2 - 1 := by omega refine le_trans (le_of_lt h₂) ?_ refine Nat.sub_le_sub_right ?_ 1 rw [add_comm] exact add_le_mul (by norm_num) gp lemma imo_2022_p5_7_7 -- (b : ℕ) (p : ℕ) -- (h₀ : 0 < b) -- (hp : Nat.Prime p) -- (hb2p : b < 2 * p) -- (h₁ : b ! ≤ (2 * p - 1)!) (gp : 2 ≤ p) : -- (gp1 : p - 1 + 1 = p) -- (f : ℕ → ℕ) -- (hf : f = fun x => x + 1) : p + 2 - 1 ≤ 2 * p - 1 := by refine Nat.sub_le_sub_right ?_ 1 rw [add_comm] exact add_le_mul (by norm_num) gp lemma imo_2022_p5_7_8 -- (b : ℕ) (p : ℕ) -- (h₀ : 0 < b) -- (hp : Nat.Prime p) -- (hb2p : b < 2 * p) -- (h₁ : b ! ≤ (2 * p - 1)!) (gp : 2 ≤ p) : -- (gp1 : p - 1 + 1 = p) -- (f : ℕ → ℕ) -- (hf : f = fun x => x + 1) : p + 2 ≤ 2 * p := by rw [add_comm] exact add_le_mul (by norm_num) gp lemma imo_2022_p5_7_9 -- (b : ℕ) (p : ℕ) -- (h₀ : 0 < b) -- (hp : Nat.Prime p) -- (hb2p : b < 2 * p) -- (h₁ : b ! ≤ (2 * p - 1)!) (gp : 2 ≤ p) (gp1 : p - 1 + 1 = p) (f : ℕ → ℕ) (hf : f = fun x => x + 1) (g₀ : Finset.prod (Finset.range (2 * p - 1)) f = Finset.prod (Finset.range (p - 1 + 1)) f * Finset.prod (Finset.Ico (p - 1 + 1) (2 * p - 1)) f) : Finset.prod (Finset.range (2 * p - 1)) f = (Finset.prod (Finset.range (p - 1)) fun x => p ^ 2 - (x + 1) ^ 2) * p := by have g₁: (Finset.range ((p - 1) + 1)).prod (fun (x : ℕ) => x + 1) = (Finset.range (p - 1)).prod (fun (x : ℕ) => x + 1) * ((p - 1) + 1) := by exact Finset.prod_range_succ _ (p - 1) rw [hf, g₁] at g₀ nth_rewrite 2 [mul_comm] at g₀ rw [← mul_assoc] at g₀ rw [gp1] at g₀ g₁ have g₂: (Finset.Ico ((p - 1) + 1) (2 * p - 1)).prod (fun (x : ℕ) => x + 1) = (Finset.range (p - 1)).prod (fun (x : ℕ) => p + (x+1)) := by rw [gp1] exact imo_2022_p5_5 p have g₃: (Finset.range (p - 1)).prod (fun (x : ℕ) => x + 1) = (Finset.range (p - 1)).prod (fun (x : ℕ) => p - (x+1)) := by exact imo_2022_p5_6 p gp rw [gp1] at g₂ rw [g₂,g₃] at g₀ have g₄: (Finset.range (p - 1)).prod (fun (x : ℕ) => p + (x+1)) * (Finset.range (p - 1)).prod (fun (x : ℕ) => p - (x+1)) = (Finset.range (p - 1)).prod (fun (x : ℕ) => (p + (x+1)) * (p - (x+1))) := by symm refine Finset.prod_mul_distrib have g₅: (fun (x : ℕ) => p ^ 2 - (x+1) ^ 2) = (fun (x : ℕ) => (p + (x+1)) * (p - (x+1))) := by ext1 x exact Nat.sq_sub_sq p (x + 1) rw [g₄,← g₅, ← hf] at g₀ exact g₀ lemma imo_2022_p5_7_10 -- (b : ℕ) (p : ℕ) -- (h₀ : 0 < b) -- (hp : Nat.Prime p) -- (hb2p : b < 2 * p) -- (h₁ : b ! ≤ (2 * p - 1)!) (gp : 2 ≤ p) (gp1 : p - 1 + 1 = p) (f : ℕ → ℕ) (hf : f = fun x => x + 1) (g₀ : Finset.prod (Finset.range (2 * p - 1)) f = Finset.prod (Finset.range (p - 1 + 1)) f * Finset.prod (Finset.Ico (p - 1 + 1) (2 * p - 1)) f) (g₁ : (Finset.prod (Finset.range (p - 1 + 1)) fun x => x + 1) = (Finset.prod (Finset.range (p - 1)) fun x => x + 1) * (p - 1 + 1)) : Finset.prod (Finset.range (2 * p - 1)) f = (Finset.prod (Finset.range (p - 1)) fun x => p ^ 2 - (x + 1) ^ 2) * p := by rw [hf, g₁] at g₀ nth_rewrite 2 [mul_comm] at g₀ rw [← mul_assoc] at g₀ rw [gp1] at g₀ g₁ have g₂: (Finset.Ico ((p - 1) + 1) (2 * p - 1)).prod (fun (x : ℕ) => x + 1) = (Finset.range (p - 1)).prod (fun (x : ℕ) => p + (x+1)) := by rw [gp1] exact imo_2022_p5_5 p have g₃: (Finset.range (p - 1)).prod (fun (x : ℕ) => x + 1) = (Finset.range (p - 1)).prod (fun (x : ℕ) => p - (x+1)) := by exact imo_2022_p5_6 p gp rw [gp1] at g₂ rw [g₂,g₃] at g₀ have g₄: (Finset.range (p - 1)).prod (fun (x : ℕ) => p + (x+1)) * (Finset.range (p - 1)).prod (fun (x : ℕ) => p - (x+1)) = (Finset.range (p - 1)).prod (fun (x : ℕ) => (p + (x+1)) * (p - (x+1))) := by symm refine Finset.prod_mul_distrib have g₅: (fun (x : ℕ) => p ^ 2 - (x+1) ^ 2) = (fun (x : ℕ) => (p + (x+1)) * (p - (x+1))) := by ext1 x exact Nat.sq_sub_sq p (x + 1) rw [g₄,← g₅, ← hf] at g₀ exact g₀ lemma imo_2022_p5_7_11 -- (b : ℕ) (p : ℕ) -- (h₀ : 0 < b) -- (hp : Nat.Prime p) -- (hb2p : b < 2 * p) -- (h₁ : b ! ≤ (2 * p - 1)!) (gp : 2 ≤ p) (gp1 : p - 1 + 1 = p) (f : ℕ → ℕ) (hf : f = fun x => x + 1) (g₀ : Finset.prod (Finset.range (2 * p - 1)) f = (Finset.prod (Finset.Ico p (2 * p - 1)) f * Finset.prod (Finset.range (p - 1)) fun x => x + 1) * p) (g₁ : (Finset.prod (Finset.range p) fun x => x + 1) = (Finset.prod (Finset.range (p - 1)) fun x => x + 1) * p) : Finset.prod (Finset.range (2 * p - 1)) f = (Finset.prod (Finset.range (p - 1)) fun x => p ^ 2 - (x + 1) ^ 2) * p := by have g₂: (Finset.Ico ((p - 1) + 1) (2 * p - 1)).prod (fun (x : ℕ) => x + 1) = (Finset.range (p - 1)).prod (fun (x : ℕ) => p + (x+1)) := by rw [gp1] exact imo_2022_p5_5 p have g₃: (Finset.range (p - 1)).prod (fun (x : ℕ) => x + 1) = (Finset.range (p - 1)).prod (fun (x : ℕ) => p - (x+1)) := by exact imo_2022_p5_6 p gp rw [gp1] at g₂ rw [hf, g₂, g₃] at g₀ have g₄: (Finset.range (p - 1)).prod (fun (x : ℕ) => p + (x+1)) * (Finset.range (p - 1)).prod (fun (x : ℕ) => p - (x+1)) = (Finset.range (p - 1)).prod (fun (x : ℕ) => (p + (x+1)) * (p - (x+1))) := by symm refine Finset.prod_mul_distrib have g₅: (fun (x : ℕ) => p ^ 2 - (x+1) ^ 2) = (fun (x : ℕ) => (p + (x+1)) * (p - (x+1))) := by ext1 x exact Nat.sq_sub_sq p (x + 1) rw [g₄,← g₅, ← hf] at g₀ exact g₀ lemma imo_2022_p5_7_12 -- (b : ℕ) (p : ℕ) -- (h₀ : 0 < b) -- (hp : Nat.Prime p) -- (hb2p : b < 2 * p) -- (h₁ : b ! ≤ (2 * p - 1)!) -- (gp : 2 ≤ p) (gp1 : p - 1 + 1 = p) (f : ℕ → ℕ) (hf : f = fun x => x + 1) (g₀ : Finset.prod (Finset.range (2 * p - 1)) f = Finset.prod (Finset.range (p - 1 + 1)) f * Finset.prod (Finset.Ico (p - 1 + 1) (2 * p - 1)) f) (g₁ : (Finset.prod (Finset.range (p - 1 + 1)) fun x => x + 1) = (Finset.prod (Finset.range (p - 1)) fun x => x + 1) * (p - 1 + 1)) : Finset.prod (Finset.range (2 * p - 1)) f = (Finset.prod (Finset.Ico p (2 * p - 1)) f * Finset.prod (Finset.range (p - 1)) fun x => x + 1) * p := by rw [hf, g₁] at g₀ nth_rewrite 2 [mul_comm] at g₀ rw [← mul_assoc] at g₀ rw [gp1] at g₀ g₁ rw [hf, g₀] lemma imo_2022_p5_7_13 -- (b : ℕ) (p : ℕ) -- (h₀ : 0 < b) -- (hp : Nat.Prime p) -- (hb2p : b < 2 * p) -- (h₁ : b ! ≤ (2 * p - 1)!) -- (gp : 2 ≤ p) (gp1 : p - 1 + 1 = p) (f : ℕ → ℕ) (hf : f = fun x => x + 1) (g₀ : Finset.prod (Finset.range (2 * p - 1)) f = (Finset.prod (Finset.Ico p (2 * p - 1)) f * Finset.prod (Finset.range (p - 1)) fun x => x + 1) * p) (g₁ : (Finset.prod (Finset.range p) fun x => x + 1) = (Finset.prod (Finset.range (p - 1)) fun x => x + 1) * p) (g₂ : (Finset.Ico ((p - 1) + 1) (2 * p - 1)).prod (fun (x : ℕ) => x + 1) = (Finset.range (p - 1)).prod (fun (x : ℕ) => p + (x+1))) (g₃ : (Finset.prod (Finset.range (p - 1)) fun x => x + 1) = Finset.prod (Finset.range (p - 1)) fun x => p - (x + 1)) : Finset.prod (Finset.range (2 * p - 1)) f = (Finset.prod (Finset.range (p - 1)) fun x => p ^ 2 - (x + 1) ^ 2) * p := by rw [gp1] at g₂ rw [hf, g₂, g₃] at g₀ have g₄: (Finset.range (p - 1)).prod (fun (x : ℕ) => p + (x+1)) * (Finset.range (p - 1)).prod (fun (x : ℕ) => p - (x+1)) = (Finset.range (p - 1)).prod (fun (x : ℕ) => (p + (x+1)) * (p - (x+1))) := by symm refine Finset.prod_mul_distrib have g₅: (fun (x : ℕ) => p ^ 2 - (x+1) ^ 2) = (fun (x : ℕ) => (p + (x+1)) * (p - (x+1))) := by ext1 x exact Nat.sq_sub_sq p (x + 1) rw [g₄,← g₅, ← hf] at g₀ exact g₀ lemma imo_2022_p5_7_14 -- (b : ℕ) (p : ℕ) -- (h₀ : 0 < b) -- (hp : Nat.Prime p) -- (hb2p : b < 2 * p) -- (h₁ : b ! ≤ (2 * p - 1)!) -- (gp : 2 ≤ p) -- (gp1 : p - 1 + 1 = p) (f : ℕ → ℕ) -- (hf : f = fun x => x + 1) (g₀ : Finset.prod (Finset.range (2 * p - 1)) f = ((Finset.prod (Finset.range (p - 1)) fun x => p + (x + 1)) * Finset.prod (Finset.range (p - 1)) fun x => p - (x + 1)) * p) (g₁ : (Finset.prod (Finset.range p) fun x => x + 1) = (Finset.prod (Finset.range (p - 1)) fun x => x + 1) * p) (g₂ : (Finset.prod (Finset.Ico p (2 * p - 1)) fun x => x + 1) = Finset.prod (Finset.range (p - 1)) fun x => p + (x + 1)) (g₃ : (Finset.prod (Finset.range (p - 1)) fun x => x + 1) = Finset.prod (Finset.range (p - 1)) fun x => p - (x + 1)) (g₄ : ((Finset.prod (Finset.range (p - 1)) fun x => p + (x + 1)) * Finset.prod (Finset.range (p - 1)) fun x => p - (x + 1)) = Finset.prod (Finset.range (p - 1)) fun x => (p + (x + 1)) * (p - (x + 1))) : Finset.prod (Finset.range (2 * p - 1)) f = (Finset.prod (Finset.range (p - 1)) fun x => p ^ 2 - (x + 1) ^ 2) * p := by have g₅: (fun (x : ℕ) => p ^ 2 - (x+1) ^ 2) = (fun (x : ℕ) => (p + (x+1)) * (p - (x+1))) := by ext1 x exact Nat.sq_sub_sq p (x + 1) rw [g₄,← g₅] at g₀ exact g₀ lemma imo_2022_p5_7_15 -- (b : ℕ) (p : ℕ) : -- (h₀ : 0 < b) -- (hp : Nat.Prime p) -- (hb2p : b < 2 * p) -- (h₁ : b ! ≤ (2 * p - 1)!) -- (gp : 2 ≤ p) -- (gp1 : p - 1 + 1 = p) -- (f : ℕ → ℕ) -- (hf : f = fun x => x + 1) -- (g₀ : Finset.prod (Finset.range (2 * p - 1)) f = -- ((Finset.prod (Finset.range (p - 1)) fun x => p + (x + 1)) * -- Finset.prod (Finset.range (p - 1)) fun x => p - (x + 1)) * p) -- (g₁ : (Finset.prod (Finset.range p) fun x => x + 1) = (Finset.prod (Finset.range (p - 1)) fun x => x + 1) * p) -- (g₂ : (Finset.prod (Finset.Ico p (2 * p - 1)) fun x => x + 1) = Finset.prod (Finset.range (p - 1)) fun x => p + (x + 1)) -- (g₃ : (Finset.prod (Finset.range (p - 1)) fun x => x + 1) = Finset.prod (Finset.range (p - 1)) fun x => p - (x + 1)) -- (g₄ : ((Finset.prod (Finset.range (p - 1)) fun x => p + (x + 1)) * Finset.prod (Finset.range (p - 1)) fun x => p - (x + 1)) = -- Finset.prod (Finset.range (p - 1)) fun x => (p + (x + 1)) * (p - (x + 1))) : (fun x => p ^ 2 - (x + 1) ^ 2) = fun x => (p + (x + 1)) * (p - (x + 1)) := by ext1 x exact Nat.sq_sub_sq p (x + 1) lemma imo_2022_p5_7_16 -- (b : ℕ) (p : ℕ) -- (h₀ : 0 < b) -- (hp : Nat.Prime p) -- (hb2p : b < 2 * p) -- (h₁ : b ! ≤ (2 * p - 1)!) -- (gp : 2 ≤ p) -- (gp1 : p - 1 + 1 = p) (f : ℕ → ℕ) -- (hf : f = fun x => x + 1) (g₀ : Finset.prod (Finset.range (2 * p - 1)) f = ((Finset.prod (Finset.range (p - 1)) fun x => p + (x + 1)) * Finset.prod (Finset.range (p - 1)) fun x => p - (x + 1)) * p) (g₁ : (Finset.prod (Finset.range p) fun x => x + 1) = (Finset.prod (Finset.range (p - 1)) fun x => x + 1) * p) (g₂ : (Finset.prod (Finset.Ico p (2 * p - 1)) fun x => x + 1) = Finset.prod (Finset.range (p - 1)) fun x => p + (x + 1)) (g₃ : (Finset.prod (Finset.range (p - 1)) fun x => x + 1) = Finset.prod (Finset.range (p - 1)) fun x => p - (x + 1)) (g₄ : ((Finset.prod (Finset.range (p - 1)) fun x => p + (x + 1)) * Finset.prod (Finset.range (p - 1)) fun x => p - (x + 1)) = Finset.prod (Finset.range (p - 1)) fun x => (p + (x + 1)) * (p - (x + 1))) (g₅ : (fun x => p ^ 2 - (x + 1) ^ 2) = fun x => (p + (x + 1)) * (p - (x + 1))) : Finset.prod (Finset.range (2 * p - 1)) f = (Finset.prod (Finset.range (p - 1)) fun x => p ^ 2 - (x + 1) ^ 2) * p := by rw [g₄,← g₅] at g₀ exact g₀ lemma imo_2022_p5_7_17 (b p : ℕ) -- (h₀ : 0 < b) -- (hp : Nat.Prime p) -- (hb2p : b < 2 * p) (h₁ : b ! ≤ (2 * p - 1)!) (gp : 2 ≤ p) (gp1 : p - 1 + 1 = p) (f : ℕ → ℕ) (hf : f = fun x => x + 1) (h₂ : Finset.prod (Finset.range (2 * p - 1)) f = (Finset.prod (Finset.range (p - 1)) fun x => p ^ 2 - (x + 1) ^ 2) * p) : b ! + p < p ^ (2 * p) := by have h₃: (Finset.range (p - 1)).prod (fun (x : ℕ) => p^2 - (x+1)^2) * p ≤ (p^2)^(Finset.range (p - 1)).card * p := by refine Nat.mul_le_mul_right ?_ ?_ refine Finset.prod_le_pow_card (Finset.range (p - 1)) ?_ (p^2) ?_ intros x _ exact (p ^ 2).sub_le ((x + 1) ^ 2) simp at * have h₄: b.factorial + p ≤ (p ^ 2) ^ (p - 1) * p + p := by refine add_le_add_right ?_ p refine le_trans ?_ h₃ rw [← h₂, hf] rw [Finset.prod_range_add_one_eq_factorial] exact h₁ have h₅: b.factorial + p < (p ^ 2) ^ (p - 1) * p * p := by refine lt_of_le_of_lt h₄ ?_ rw [add_comm] nth_rewrite 2 [mul_comm] refine imo_2022_p5_4 p ((p ^ 2) ^ (p - 1) * p) gp ?_ refine lt_mul_left (by linarith) ?_ rw [← pow_mul] refine Nat.one_lt_pow ?_ (Nat.lt_of_succ_le gp) refine Nat.mul_ne_zero (by norm_num) ?_ exact Nat.sub_ne_zero_iff_lt.mpr gp rw [mul_assoc _ p p, ← pow_two p] at h₅ rw [← Nat.pow_succ, succ_eq_add_one, gp1] at h₅ rw [Nat.pow_mul] exact h₅ lemma imo_2022_p5_7_18 -- (b : ℕ) (p : ℕ) : -- (h₀ : 0 < b) -- (hp : Nat.Prime p) -- (hb2p : b < 2 * p) -- (h₁ : b ! ≤ (2 * p - 1)!) -- (gp : 2 ≤ p) -- (gp1 : p - 1 + 1 = p) -- (f : ℕ → ℕ) -- (hf : f = fun x => x + 1) -- (h₂ : Finset.prod (Finset.range (2 * p - 1)) f = (Finset.prod (Finset.range (p - 1)) fun x => p ^ 2 - (x + 1) ^ 2) * p) : (Finset.prod (Finset.range (p - 1)) fun x => p ^ 2 - (x + 1) ^ 2) * p ≤ (p ^ 2) ^ (Finset.range (p - 1)).card * p := by refine Nat.mul_le_mul_right ?_ ?_ refine Finset.prod_le_pow_card (Finset.range (p - 1)) ?_ (p^2) ?_ intros x _ exact (p ^ 2).sub_le ((x + 1) ^ 2) lemma imo_2022_p5_7_19 -- (b : ℕ) (p : ℕ) : -- (h₀ : 0 < b) -- (hp : Nat.Prime p) -- (hb2p : b < 2 * p) -- (h₁ : b ! ≤ (2 * p - 1)!) -- (gp : 2 ≤ p) -- (gp1 : p - 1 + 1 = p) -- (f : ℕ → ℕ) -- (hf : f = fun x => x + 1) -- (h₂ : Finset.prod (Finset.range (2 * p - 1)) f = (Finset.prod (Finset.range (p - 1)) fun x => p ^ 2 - (x + 1) ^ 2) * p) : (Finset.prod (Finset.range (p - 1)) fun x => (p ^ 2 - (x + 1) ^ 2)) ≤ (p ^ 2) ^ (Finset.range (p - 1)).card := by refine Finset.prod_le_pow_card (Finset.range (p - 1)) ?_ (p^2) ?_ intros x _ exact (p ^ 2).sub_le ((x + 1) ^ 2) lemma imo_2022_p5_7_20 -- (b : ℕ) (p : ℕ) : -- (h₀ : 0 < b) -- (hp : Nat.Prime p) -- (hb2p : b < 2 * p) -- (h₁ : b ! ≤ (2 * p - 1)!) -- (gp : 2 ≤ p) -- (gp1 : p - 1 + 1 = p) -- (f : ℕ → ℕ) -- (hf : f = fun x => x + 1) -- (h₂ : Finset.prod (Finset.range (2 * p - 1)) f = (Finset.prod (Finset.range (p - 1)) fun x => p ^ 2 - (x + 1) ^ 2) * p) : ∀ x ∈ Finset.range (p - 1), p ^ 2 - (x + 1) ^ 2 ≤ p ^ 2 := by intros x _ exact (p ^ 2).sub_le ((x + 1) ^ 2) lemma imo_2022_p5_7_21 (b p : ℕ) -- (h₀ : 0 < b) -- (hp : Nat.Prime p) -- (hb2p : b < 2 * p) (h₁ : b ! ≤ (2 * p - 1)!) (gp : 2 ≤ p) (gp1 : p - 1 + 1 = p) (f : ℕ → ℕ) (hf : f = fun x => x + 1) (h₂ : Finset.prod (Finset.range (2 * p - 1)) f = (Finset.prod (Finset.range (p - 1)) fun x => p ^ 2 - (x + 1) ^ 2) * p) (h₃ : (Finset.prod (Finset.range (p - 1)) fun x => p ^ 2 - (x + 1) ^ 2) * p ≤ (p ^ 2) ^ (Finset.range (p - 1)).card * p) : b ! + p < p ^ (2 * p) := by simp at * have h₄: b.factorial + p ≤ (p ^ 2) ^ (p - 1) * p + p := by refine add_le_add_right ?_ p refine le_trans ?_ h₃ rw [← h₂, hf] rw [Finset.prod_range_add_one_eq_factorial] exact h₁ have h₅: b.factorial + p < (p ^ 2) ^ (p - 1) * p * p := by refine lt_of_le_of_lt h₄ ?_ rw [add_comm] nth_rewrite 2 [mul_comm] refine imo_2022_p5_4 p ((p ^ 2) ^ (p - 1) * p) gp ?_ refine lt_mul_left (by linarith) ?_ rw [← pow_mul] refine Nat.one_lt_pow ?_ (Nat.lt_of_succ_le gp) refine Nat.mul_ne_zero (by norm_num) ?_ exact Nat.sub_ne_zero_iff_lt.mpr gp rw [mul_assoc _ p p, ← pow_two p] at h₅ rw [← Nat.pow_succ, succ_eq_add_one, gp1] at h₅ rw [Nat.pow_mul] exact h₅ lemma imo_2022_p5_7_22 (b p : ℕ) -- (h₀ : 0 < b) -- (hp : Nat.Prime p) -- (hb2p : b < 2 * p) (h₁ : b ! ≤ (2 * p - 1)!) -- (gp : 2 ≤ p) -- (gp1 : p - 1 + 1 = p) (f : ℕ → ℕ) (hf : f = fun x => x + 1) (h₂ : Finset.prod (Finset.range (2 * p - 1)) f = (Finset.prod (Finset.range (p - 1)) fun x => p ^ 2 - (x + 1) ^ 2) * p) (h₃ : (Finset.prod (Finset.range (p - 1)) fun x => p ^ 2 - (x + 1) ^ 2) * p ≤ (p ^ 2) ^ (p - 1) * p) : b ! + p ≤ (p ^ 2) ^ (p - 1) * p + p := by refine add_le_add_right ?_ p refine le_trans ?_ h₃ rw [← h₂, hf] rw [Finset.prod_range_add_one_eq_factorial] exact h₁ lemma imo_2022_p5_7_23 (b p : ℕ) -- (h₀ : 0 < b) -- (hp : Nat.Prime p) -- (hb2p : b < 2 * p) (h₁ : b ! ≤ (2 * p - 1)!) -- (gp : 2 ≤ p) -- (gp1 : p - 1 + 1 = p) (f : ℕ → ℕ) (hf : f = fun x => x + 1) (h₂ : Finset.prod (Finset.range (2 * p - 1)) f = (Finset.prod (Finset.range (p - 1)) fun x => p ^ 2 - (x + 1) ^ 2) * p) (h₃ : (Finset.prod (Finset.range (p - 1)) fun x => p ^ 2 - (x + 1) ^ 2) * p ≤ (p ^ 2) ^ (p - 1) * p) : b ! ≤ (p ^ 2) ^ (p - 1) * p := by refine le_trans ?_ h₃ rw [← h₂, hf] rw [Finset.prod_range_add_one_eq_factorial] exact h₁ lemma imo_2022_p5_7_24 (b p : ℕ) -- (h₀ : 0 < b) -- (hp : Nat.Prime p) -- (hb2p : b < 2 * p) (h₁ : b ! ≤ (2 * p - 1)!) -- (gp : 2 ≤ p) -- (gp1 : p - 1 + 1 = p) (f : ℕ → ℕ) (hf : f = fun x => x + 1) : -- (h₂ : Finset.prod (Finset.range (2 * p - 1)) f = (Finset.prod (Finset.range (p - 1)) fun x => p ^ 2 - (x + 1) ^ 2) * p) -- (h₃ : (Finset.prod (Finset.range (p - 1)) fun x => p ^ 2 - (x + 1) ^ 2) * p ≤ (p ^ 2) ^ (p - 1) * p) : b ! ≤ Finset.prod (Finset.range (2 * p - 1)) f := by rw [hf] rw [Finset.prod_range_add_one_eq_factorial] exact h₁ lemma imo_2022_p5_7_25 (b p : ℕ) -- (h₀ : 0 < b) -- (hp : Nat.Prime p) -- (hb2p : b < 2 * p) -- (h₁ : b ! ≤ (2 * p - 1)!) (gp : 2 ≤ p) (gp1 : p - 1 + 1 = p) -- (f : ℕ → ℕ) -- (hf : f = fun x => x + 1) -- (h₂ : Finset.prod (Finset.range (2 * p - 1)) f = (Finset.prod (Finset.range (p - 1)) fun x => p ^ 2 - (x + 1) ^ 2) * p) -- (h₃ : (Finset.prod (Finset.range (p - 1)) fun x => p ^ 2 - (x + 1) ^ 2) * p ≤ (p ^ 2) ^ (p - 1) * p) (h₄ : b ! + p ≤ (p ^ 2) ^ (p - 1) * p + p) : b ! + p < p ^ (2 * p) := by have h₅: b.factorial + p < (p ^ 2) ^ (p - 1) * p * p := by refine lt_of_le_of_lt h₄ ?_ rw [add_comm] nth_rewrite 2 [mul_comm] refine imo_2022_p5_4 p ((p ^ 2) ^ (p - 1) * p) gp ?_ refine lt_mul_left (by linarith) ?_ rw [← pow_mul] refine Nat.one_lt_pow ?_ (Nat.lt_of_succ_le gp) refine Nat.mul_ne_zero (by norm_num) ?_ exact Nat.sub_ne_zero_iff_lt.mpr gp rw [mul_assoc _ p p, ← pow_two p] at h₅ rw [← Nat.pow_succ, succ_eq_add_one, gp1] at h₅ rw [Nat.pow_mul] exact h₅ lemma imo_2022_p5_7_26 (b p : ℕ) -- (h₀ : 0 < b) -- (hp : Nat.Prime p) -- (hb2p : b < 2 * p) -- (h₁ : b ! ≤ (2 * p - 1)!) (gp : 2 ≤ p) -- (gp1 : p - 1 + 1 = p) -- (f : ℕ → ℕ) -- (hf : f = fun x => x + 1) -- (h₂ : Finset.prod (Finset.range (2 * p - 1)) f = (Finset.prod (Finset.range (p - 1)) fun x => p ^ 2 - (x + 1) ^ 2) * p) -- (h₃ : (Finset.prod (Finset.range (p - 1)) fun x => p ^ 2 - (x + 1) ^ 2) * p ≤ (p ^ 2) ^ (p - 1) * p) (h₄ : b ! + p ≤ (p ^ 2) ^ (p - 1) * p + p) : b ! + p < (p ^ 2) ^ (p - 1) * p * p := by refine lt_of_le_of_lt h₄ ?_ rw [add_comm] nth_rewrite 2 [mul_comm] refine imo_2022_p5_4 p ((p ^ 2) ^ (p - 1) * p) gp ?_ refine lt_mul_left (by linarith) ?_ rw [← pow_mul] refine Nat.one_lt_pow ?_ (Nat.lt_of_succ_le gp) refine Nat.mul_ne_zero (by norm_num) ?_ exact Nat.sub_ne_zero_iff_lt.mpr gp lemma imo_2022_p5_7_27 (p : ℕ) -- (h₀ : 0 < b) -- (hp : Nat.Prime p) -- (hb2p : b < 2 * p) -- (h₁ : b ! ≤ (2 * p - 1)!) (gp : 2 ≤ p) : -- (gp1 : p - 1 + 1 = p) -- (f : ℕ → ℕ) -- (hf : f = fun x => x + 1) -- (h₂ : Finset.prod (Finset.range (2 * p - 1)) f = (Finset.prod (Finset.range (p - 1)) fun x => p ^ 2 - (x + 1) ^ 2) * p) -- (h₃ : (Finset.prod (Finset.range (p - 1)) fun x => p ^ 2 - (x + 1) ^ 2) * p ≤ (p ^ 2) ^ (p - 1) * p) -- (h₄ : b ! + p ≤ (p ^ 2) ^ (p - 1) * p + p) : (p ^ 2) ^ (p - 1) * p + p < (p ^ 2) ^ (p - 1) * p * p := by rw [add_comm] nth_rewrite 2 [mul_comm] refine imo_2022_p5_4 p ((p ^ 2) ^ (p - 1) * p) gp ?_ refine lt_mul_left (by linarith) ?_ rw [← pow_mul] refine Nat.one_lt_pow ?_ (Nat.lt_of_succ_le gp) refine Nat.mul_ne_zero (by norm_num) ?_ exact Nat.sub_ne_zero_iff_lt.mpr gp lemma imo_2022_p5_7_28 (p : ℕ) -- (h₀ : 0 < b) -- (hp : Nat.Prime p) -- (hb2p : b < 2 * p) -- (h₁ : b ! ≤ (2 * p - 1)!) (gp : 2 ≤ p) : -- (gp1 : p - 1 + 1 = p) -- (f : ℕ → ℕ) -- (hf : f = fun x => x + 1) -- (h₂ : Finset.prod (Finset.range (2 * p - 1)) f = (Finset.prod (Finset.range (p - 1)) fun x => p ^ 2 - (x + 1) ^ 2) * p) -- (h₃ : (Finset.prod (Finset.range (p - 1)) fun x => p ^ 2 - (x + 1) ^ 2) * p ≤ (p ^ 2) ^ (p - 1) * p) -- (h₄ : b ! + p ≤ (p ^ 2) ^ (p - 1) * p + p) : p < (p ^ 2) ^ (p - 1) * p := by refine lt_mul_left (by linarith) ?_ rw [← pow_mul] refine Nat.one_lt_pow ?_ (Nat.lt_of_succ_le gp) refine Nat.mul_ne_zero (by norm_num) ?_ exact Nat.sub_ne_zero_iff_lt.mpr gp lemma imo_2022_p5_7_29 -- (b : ℕ) (p : ℕ) -- (h₀ : 0 < b) -- (hp : Nat.Prime p) -- (hb2p : b < 2 * p) -- (h₁ : b ! ≤ (2 * p - 1)!) (gp : 2 ≤ p) : -- (gp1 : p - 1 + 1 = p) -- (f : ℕ → ℕ) -- (hf : f = fun x => x + 1) -- (h₂ : Finset.prod (Finset.range (2 * p - 1)) f = (Finset.prod (Finset.range (p - 1)) fun x => p ^ 2 - (x + 1) ^ 2) * p) -- (h₃ : (Finset.prod (Finset.range (p - 1)) fun x => p ^ 2 - (x + 1) ^ 2) * p ≤ (p ^ 2) ^ (p - 1) * p) -- (h₄ : b ! + p ≤ (p ^ 2) ^ (p - 1) * p + p) : 1 < p ^ (2 * (p - 1)) := by refine Nat.one_lt_pow ?_ (Nat.lt_of_succ_le gp) refine Nat.mul_ne_zero (by norm_num) ?_ exact Nat.sub_ne_zero_iff_lt.mpr gp lemma imo_2022_p5_7_30 -- (b : ℕ) (p : ℕ) -- (h₀ : 0 < b) -- (hp : Nat.Prime p) -- (hb2p : b < 2 * p) -- (h₁ : b ! ≤ (2 * p - 1)!) (gp : 2 ≤ p) : -- (gp1 : p - 1 + 1 = p) -- (f : ℕ → ℕ) -- (hf : f = fun x => x + 1) -- (h₂ : Finset.prod (Finset.range (2 * p - 1)) f = (Finset.prod (Finset.range (p - 1)) fun x => p ^ 2 - (x + 1) ^ 2) * p) -- (h₃ : (Finset.prod (Finset.range (p - 1)) fun x => p ^ 2 - (x + 1) ^ 2) * p ≤ (p ^ 2) ^ (p - 1) * p) -- (h₄ : b ! + p ≤ (p ^ 2) ^ (p - 1) * p + p) : 2 * (p - 1) ≠ 0 := by refine Nat.mul_ne_zero (by norm_num) ?_ exact Nat.sub_ne_zero_iff_lt.mpr gp lemma imo_2022_p5_7_31 (b p : ℕ) -- (h₀ : 0 < b) -- (hp : Nat.Prime p) -- (hb2p : b < 2 * p) -- (h₁ : b ! ≤ (2 * p - 1)!) -- (gp : 2 ≤ p) (gp1 : p - 1 + 1 = p) -- (f : ℕ → ℕ) -- (hf : f = fun x => x + 1) -- (h₂ : Finset.prod (Finset.range (2 * p - 1)) f = (Finset.prod (Finset.range (p - 1)) fun x => p ^ 2 - (x + 1) ^ 2) * p) -- (h₃ : (Finset.prod (Finset.range (p - 1)) fun x => p ^ 2 - (x + 1) ^ 2) * p ≤ (p ^ 2) ^ (p - 1) * p) -- (h₄ : b ! + p ≤ (p ^ 2) ^ (p - 1) * p + p) (h₅ : b ! + p < (p ^ 2) ^ (p - 1) * p * p) : b ! + p < p ^ (2 * p) := by rw [mul_assoc _ p p, ← pow_two p] at h₅ rw [← Nat.pow_succ, succ_eq_add_one, gp1] at h₅ rw [Nat.pow_mul] exact h₅ lemma imo_2022_p5_7_32 (b p : ℕ) -- (h₀ : 0 < b) -- (hp : Nat.Prime p) -- (hb2p : b < 2 * p) -- (h₁ : b ! ≤ (2 * p - 1)!) -- (gp : 2 ≤ p) -- (gp1 : p - 1 + 1 = p) -- (f : ℕ → ℕ) -- (hf : f = fun x => x + 1) -- (h₂ : Finset.prod (Finset.range (2 * p - 1)) f = (Finset.prod (Finset.range (p - 1)) fun x => p ^ 2 - (x + 1) ^ 2) * p) -- (h₃ : (Finset.prod (Finset.range (p - 1)) fun x => p ^ 2 - (x + 1) ^ 2) * p ≤ (p ^ 2) ^ (p - 1) * p) -- (h₄ : b ! + p ≤ (p ^ 2) ^ (p - 1) * p + p) (h₅ : b ! + p < (p ^ 2) ^ p) : b ! + p < p ^ (2 * p) := by rw [Nat.pow_mul] exact h₅ lemma imo_2022_p5_7_33 (b p : ℕ) -- (h₀ : 0 < b) -- (hp : Nat.Prime p) -- (hb2p : b < 2 * p) -- (h₁ : b ! ≤ (2 * p - 1)!) -- (gp : 2 ≤ p) (gp1 : p - 1 + 1 = p) -- (f : ℕ → ℕ) -- (hf : f = fun x => x + 1) -- (h₂ : Finset.prod (Finset.range (2 * p - 1)) f = (Finset.prod (Finset.range (p - 1)) fun x => p ^ 2 - (x + 1) ^ 2) * p) -- (h₃ : (Finset.prod (Finset.range (p - 1)) fun x => p ^ 2 - (x + 1) ^ 2) * p ≤ (p ^ 2) ^ (p - 1) * p) -- (h₄ : b ! + p ≤ (p ^ 2) ^ (p - 1) * p + p) (h₅ : b ! + p < (p ^ 2) ^ (p - 1) * p * p) : b ! + p < (p ^ 2) ^ p := by rw [mul_assoc _ p p, ← pow_two p] at h₅ rw [← Nat.pow_succ, succ_eq_add_one, gp1] at h₅ exact h₅ lemma imo_2022_p5_8 (a b p: ℕ) (h₀: 0 < a ∧ 0 < b) (hp: Nat.Prime p) (h₁: a ^ p = b.factorial + p) (hbp: p ≤ b) (h₂: p ∣ a) (hb2p: b < 2 * p) : (a = p) := by have gp: p ≤ a := by exact Nat.le_of_dvd h₀.1 h₂ cases' lt_or_eq_of_le gp with h₃ h₃ . exfalso cases' h₂ with c h₂ have gc: 0 < c := by by_contra hc0 push_neg at hc0 interval_cases c simp at * linarith by_cases hc: c < p . have g₁: c ∣ c^p := by exact dvd_pow_self c (by linarith) have h₄: c ∣ a^p := by rw [h₂, mul_pow] exact dvd_mul_of_dvd_right g₁ (p ^ p) have h₅: c ∣ b.factorial := by exact Nat.dvd_factorial gc (by linarith) have g₂: p = a ^ p - b.factorial := by symm rw [add_comm] at h₁ refine (Nat.sub_eq_iff_eq_add ?_).mpr h₁ rw [add_comm] at h₁ exact le.intro (h₁.symm) have h₆: c ∣ p := by rw [g₂] exact dvd_sub' h₄ h₅ have h₇: c = 1 ∨ c = p := by exact (dvd_prime hp).mp h₆ cases' h₇ with h₇₀ h₇₁ . rw [h₇₀, mul_one] at h₂ rw [h₂] at h₃ linarith [h₃] . rw [h₇₁] at hc simp at hc . push_neg at hc have g₃: p^2 ≤ a := by rw [h₂, pow_two] exact mul_le_mul_left' hc p have h₃: p^(2*p) ≤ a^p := by rw [pow_mul] exact pow_left_mono p g₃ have h₇: b.factorial + p < p^(2*p) := by exact imo_2022_p5_7 b p hp hb2p rw [←h₁] at h₇ linarith exact h₃.symm lemma imo_2022_p5_8_1 (a b p : ℕ) (h₀ : 0 < a ∧ 0 < b) (hp : Nat.Prime p) (h₁ : a ^ p = b ! + p) (hbp : p ≤ b) (h₂ : p ∣ a) (hb2p : b < 2 * p) (gp : p ≤ a) : a = p := by cases' lt_or_eq_of_le gp with h₃ h₃ . exfalso cases' h₂ with c h₂ have gc: 0 < c := by by_contra hc0 push_neg at hc0 interval_cases c simp at * linarith by_cases hc: c < p . have g₁: c ∣ c^p := by exact dvd_pow_self c (by linarith) have h₄: c ∣ a^p := by rw [h₂, mul_pow] exact dvd_mul_of_dvd_right g₁ (p ^ p) have h₅: c ∣ b.factorial := by exact Nat.dvd_factorial gc (by linarith) have g₂: p = a ^ p - b.factorial := by symm rw [add_comm] at h₁ refine (Nat.sub_eq_iff_eq_add ?_).mpr h₁ rw [add_comm] at h₁ exact le.intro (h₁.symm) have h₆: c ∣ p := by rw [g₂] exact dvd_sub' h₄ h₅ have h₇: c = 1 ∨ c = p := by exact (dvd_prime hp).mp h₆ cases' h₇ with h₇₀ h₇₁ . rw [h₇₀, mul_one] at h₂ rw [h₂] at h₃ linarith [h₃] . rw [h₇₁] at hc simp at hc . push_neg at hc have g₃: p^2 ≤ a := by rw [h₂, pow_two] exact mul_le_mul_left' hc p have h₃: p^(2*p) ≤ a^p := by rw [pow_mul] exact pow_left_mono p g₃ have h₇: b.factorial + p < p^(2*p) := by exact imo_2022_p5_7 b p hp hb2p rw [←h₁] at h₇ linarith . exact h₃.symm lemma imo_2022_p5_8_2 (a b p : ℕ) (h₀ : 0 < a ∧ 0 < b) (hp : Nat.Prime p) (h₁ : a ^ p = b ! + p) (hbp : p ≤ b) (h₂ : p ∣ a) (hb2p : b < 2 * p) -- (gp : p ≤ a) (h₃ : p < a) : a = p := by exfalso cases' h₂ with c h₂ have gc: 0 < c := by by_contra hc0 push_neg at hc0 interval_cases c simp at * linarith by_cases hc: c < p . have g₁: c ∣ c^p := by exact dvd_pow_self c (by linarith) have h₄: c ∣ a^p := by rw [h₂, mul_pow] exact dvd_mul_of_dvd_right g₁ (p ^ p) have h₅: c ∣ b.factorial := by exact Nat.dvd_factorial gc (by linarith) have g₂: p = a ^ p - b.factorial := by symm rw [add_comm] at h₁ refine (Nat.sub_eq_iff_eq_add ?_).mpr h₁ rw [add_comm] at h₁ exact le.intro (h₁.symm) have h₆: c ∣ p := by rw [g₂] exact dvd_sub' h₄ h₅ have h₇: c = 1 ∨ c = p := by exact (dvd_prime hp).mp h₆ cases' h₇ with h₇₀ h₇₁ . rw [h₇₀, mul_one] at h₂ rw [h₂] at h₃ linarith [h₃] . rw [h₇₁] at hc simp at hc . push_neg at hc have g₃: p^2 ≤ a := by rw [h₂, pow_two] exact mul_le_mul_left' hc p have h₃: p^(2*p) ≤ a^p := by rw [pow_mul] exact pow_left_mono p g₃ have h₇: b.factorial + p < p^(2*p) := by exact imo_2022_p5_7 b p hp hb2p rw [←h₁] at h₇ linarith lemma imo_2022_p5_8_3 (a b p : ℕ) (h₀ : 0 < a ∧ 0 < b) (hp : Nat.Prime p) (h₁ : a ^ p = b ! + p) (hbp : p ≤ b) (hb2p : b < 2 * p) -- (gp : p ≤ a) (h₃ : p < a) (c : ℕ) (h₂ : a = p * c) : False := by have gc: 0 < c := by by_contra hc0 push_neg at hc0 interval_cases c simp at * linarith by_cases hc: c < p . have g₁: c ∣ c^p := by exact dvd_pow_self c (by linarith) have h₄: c ∣ a^p := by rw [h₂, mul_pow] exact dvd_mul_of_dvd_right g₁ (p ^ p) have h₅: c ∣ b.factorial := by exact Nat.dvd_factorial gc (by linarith) have g₂: p = a ^ p - b.factorial := by symm rw [add_comm] at h₁ refine (Nat.sub_eq_iff_eq_add ?_).mpr h₁ rw [add_comm] at h₁ exact le.intro (h₁.symm) have h₆: c ∣ p := by rw [g₂] exact dvd_sub' h₄ h₅ have h₇: c = 1 ∨ c = p := by exact (dvd_prime hp).mp h₆ cases' h₇ with h₇₀ h₇₁ . rw [h₇₀, mul_one] at h₂ rw [h₂] at h₃ linarith [h₃] . rw [h₇₁] at hc simp at hc . push_neg at hc have g₃: p^2 ≤ a := by rw [h₂, pow_two] exact mul_le_mul_left' hc p have h₃: p^(2*p) ≤ a^p := by rw [pow_mul] exact pow_left_mono p g₃ have h₇: b.factorial + p < p^(2*p) := by exact imo_2022_p5_7 b p hp hb2p rw [←h₁] at h₇ linarith lemma imo_2022_p5_8_4 (a b p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) -- (hp : Nat.Prime p) -- (h₁ : a ^ p = b ! + p) (hbp : p ≤ b) (hb2p : b < 2 * p) -- (gp : p ≤ a) (h₃ : p < a) (c : ℕ) (h₂ : a = p * c) : 0 < c := by by_contra hc0 push_neg at hc0 interval_cases c simp at * linarith lemma imo_2022_p5_8_5 (a b p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) -- (hp : Nat.Prime p) -- (h₁ : a ^ p = b ! + p) (hbp : p ≤ b) (hb2p : b < 2 * p) -- (gp : p ≤ a) (h₃ : p < a) (c : ℕ) (h₂ : a = p * c) (hc0 : c ≤ 0) : False := by interval_cases c simp at * linarith lemma imo_2022_p5_8_6 (a b p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) -- (hp : Nat.Prime p) -- (h₁ : a ^ p = b ! + p) (hbp : p ≤ b) (hb2p : b < 2 * p) -- (gp : p ≤ a) (h₃ : p < a) -- (c : ℕ) (h₂ : a = p * 0) (hc0 : 0 ≤ 0) : False := by simp at * linarith lemma imo_2022_p5_8_7 (a b p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) (hp : Nat.Prime p) (h₁ : a ^ p = b ! + p) (hbp : p ≤ b) (hb2p : b < 2 * p) -- (gp : p ≤ a) (h₃ : p < a) (c : ℕ) (h₂ : a = p * c) (gc : 0 < c) : False := by by_cases hc: c < p . have g₁: c ∣ c^p := by exact dvd_pow_self c (by linarith) have h₄: c ∣ a^p := by rw [h₂, mul_pow] exact dvd_mul_of_dvd_right g₁ (p ^ p) have h₅: c ∣ b.factorial := by exact Nat.dvd_factorial gc (by linarith) have g₂: p = a ^ p - b.factorial := by symm rw [add_comm] at h₁ refine (Nat.sub_eq_iff_eq_add ?_).mpr h₁ rw [add_comm] at h₁ exact le.intro (h₁.symm) have h₆: c ∣ p := by rw [g₂] exact dvd_sub' h₄ h₅ have h₇: c = 1 ∨ c = p := by exact (dvd_prime hp).mp h₆ cases' h₇ with h₇₀ h₇₁ . rw [h₇₀, mul_one] at h₂ rw [h₂] at h₃ linarith [h₃] . rw [h₇₁] at hc simp at hc . push_neg at hc have g₃: p^2 ≤ a := by rw [h₂, pow_two] exact mul_le_mul_left' hc p have h₃: p^(2*p) ≤ a^p := by rw [pow_mul] exact pow_left_mono p g₃ have h₇: b.factorial + p < p^(2*p) := by exact imo_2022_p5_7 b p hp hb2p rw [←h₁] at h₇ linarith lemma imo_2022_p5_8_8 (a b p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) (hp : Nat.Prime p) (h₁ : a ^ p = b ! + p) (hbp : p ≤ b) -- (hb2p : b < 2 * p) -- (gp : p ≤ a) (h₃ : p < a) (c : ℕ) (h₂ : a = p * c) (gc : 0 < c) (hc : c < p) : False := by have g₁: c ∣ c^p := by exact dvd_pow_self c (by linarith) have h₄: c ∣ a^p := by rw [h₂, mul_pow] exact dvd_mul_of_dvd_right g₁ (p ^ p) have h₅: c ∣ b.factorial := by exact Nat.dvd_factorial gc (by linarith) have g₂: p = a ^ p - b.factorial := by symm rw [add_comm] at h₁ refine (Nat.sub_eq_iff_eq_add ?_).mpr h₁ rw [add_comm] at h₁ exact le.intro (h₁.symm) have h₆: c ∣ p := by rw [g₂] exact dvd_sub' h₄ h₅ have h₇: c = 1 ∨ c = p := by exact (dvd_prime hp).mp h₆ cases' h₇ with h₇₀ h₇₁ . rw [h₇₀, mul_one] at h₂ rw [h₂] at h₃ linarith [h₃] . rw [h₇₁] at hc simp at hc lemma imo_2022_p5_8_9 (a b p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) (hp : Nat.Prime p) (h₁ : a ^ p = b ! + p) (hbp : p ≤ b) -- (hb2p : b < 2 * p) -- (gp : p ≤ a) (h₃ : p < a) (c : ℕ) (h₂ : a = p * c) (gc : 0 < c) (hc : c < p) (g₁ : c ∣ c ^ p) : False := by have h₄: c ∣ a^p := by rw [h₂, mul_pow] exact dvd_mul_of_dvd_right g₁ (p ^ p) have h₅: c ∣ b.factorial := by exact Nat.dvd_factorial gc (by linarith) have g₂: p = a ^ p - b.factorial := by symm rw [add_comm] at h₁ refine (Nat.sub_eq_iff_eq_add ?_).mpr h₁ rw [add_comm] at h₁ exact le.intro (h₁.symm) have h₆: c ∣ p := by rw [g₂] exact dvd_sub' h₄ h₅ have h₇: c = 1 ∨ c = p := by exact (dvd_prime hp).mp h₆ cases' h₇ with h₇₀ h₇₁ . rw [h₇₀, mul_one] at h₂ rw [h₂] at h₃ linarith [h₃] . rw [h₇₁] at hc simp at hc lemma imo_2022_p5_8_10 (a p : ℕ) -- (b : ℕ) -- (h₀ : 0 < a ∧ 0 < b) -- (hp : Nat.Prime p) -- (h₁ : a ^ p = b ! + p) -- (hbp : p ≤ b) -- (hb2p : b < 2 * p) -- (gp : p ≤ a) -- (h₃ : p < a) (c : ℕ) (h₂ : a = p * c) -- (gc : 0 < c) -- (hc : c < p) (g₁ : c ∣ c ^ p) : c ∣ a ^ p := by rw [h₂, mul_pow] exact dvd_mul_of_dvd_right g₁ (p ^ p) lemma imo_2022_p5_8_11 (a b p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) (hp : Nat.Prime p) (h₁ : a ^ p = b ! + p) (hbp : p ≤ b) -- (hb2p : b < 2 * p) -- (gp : p ≤ a) (h₃ : p < a) (c : ℕ) (h₂ : a = p * c) (gc : 0 < c) (hc : c < p) (g₁ : c ∣ c ^ p) (h₄ : c ∣ a ^ p) : False := by have h₅: c ∣ b.factorial := by exact Nat.dvd_factorial gc (by linarith) have g₂: p = a ^ p - b.factorial := by symm rw [add_comm] at h₁ refine (Nat.sub_eq_iff_eq_add ?_).mpr h₁ rw [add_comm] at h₁ exact le.intro (h₁.symm) have h₆: c ∣ p := by rw [g₂] exact dvd_sub' h₄ h₅ have h₇: c = 1 ∨ c = p := by exact (dvd_prime hp).mp h₆ cases' h₇ with h₇₀ h₇₁ . rw [h₇₀, mul_one] at h₂ rw [h₂] at h₃ linarith [h₃] . rw [h₇₁] at hc simp at hc lemma imo_2022_p5_8_12 (a b p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) (hp : Nat.Prime p) (h₁ : a ^ p = b ! + p) -- (hbp : p ≤ b) -- (hb2p : b < 2 * p) -- (gp : p ≤ a) (h₃ : p < a) (c : ℕ) (h₂ : a = p * c) (gc : 0 < c) (hc : c < p) (g₁ : c ∣ c ^ p) (h₄ : c ∣ a ^ p) (h₅ : c ∣ b !) : False := by have g₂: p = a ^ p - b.factorial := by symm rw [add_comm] at h₁ refine (Nat.sub_eq_iff_eq_add ?_).mpr h₁ rw [add_comm] at h₁ exact le.intro (h₁.symm) have h₆: c ∣ p := by rw [g₂] exact dvd_sub' h₄ h₅ have h₇: c = 1 ∨ c = p := by exact (dvd_prime hp).mp h₆ cases' h₇ with h₇₀ h₇₁ . rw [h₇₀, mul_one] at h₂ rw [h₂] at h₃ linarith [h₃] . rw [h₇₁] at hc simp at hc lemma imo_2022_p5_8_13 (a b p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) -- (hp : Nat.Prime p) (h₁ : a ^ p = b ! + p) (hbp : p ≤ b) (hb2p : b < 2 * p) (gp : p ≤ a) (h₃ : p < a) (c : ℕ) (h₂ : a = p * c) (gc : 0 < c) (hc : c < p) (g₁ : c ∣ c ^ p) (h₄ : c ∣ a ^ p) (h₅ : c ∣ b !) : p = a ^ p - b ! := by symm rw [add_comm] at h₁ refine (Nat.sub_eq_iff_eq_add ?_).mpr h₁ rw [add_comm] at h₁ exact le.intro (h₁.symm) lemma imo_2022_p5_8_14 (a b p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) -- (hp : Nat.Prime p) (h₁ : a ^ p = p + b !) (hbp : p ≤ b) (hb2p : b < 2 * p) (gp : p ≤ a) (h₃ : p < a) (c : ℕ) (h₂ : a = p * c) (gc : 0 < c) (hc : c < p) (g₁ : c ∣ c ^ p) (h₄ : c ∣ a ^ p) (h₅ : c ∣ b !) : a ^ p - b ! = p := by refine (Nat.sub_eq_iff_eq_add ?_).mpr h₁ rw [add_comm] at h₁ exact le.intro (h₁.symm) lemma imo_2022_p5_8_15 (a b p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) -- (hp : Nat.Prime p) (h₁ : a ^ p = p + b !) (hbp : p ≤ b) (hb2p : b < 2 * p) (gp : p ≤ a) (h₃ : p < a) (c : ℕ) (h₂ : a = p * c) (gc : 0 < c) (hc : c < p) (g₁ : c ∣ c ^ p) (h₄ : c ∣ a ^ p) (h₅ : c ∣ b !) : b ! ≤ a ^ p := by rw [add_comm] at h₁ exact le.intro (h₁.symm) lemma imo_2022_p5_8_16 (a b p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) (hp : Nat.Prime p) -- (h₁ : a ^ p = b ! + p) -- (hbp : p ≤ b) -- (hb2p : b < 2 * p) -- (gp : p ≤ a) (h₃ : p < a) (c : ℕ) (h₂ : a = p * c) (gc : 0 < c) (hc : c < p) (g₁ : c ∣ c ^ p) (h₄ : c ∣ a ^ p) (h₅ : c ∣ b !) (g₂ : p = a ^ p - b !) : False := by have h₆: c ∣ p := by rw [g₂] exact dvd_sub' h₄ h₅ have h₇: c = 1 ∨ c = p := by exact (dvd_prime hp).mp h₆ cases' h₇ with h₇₀ h₇₁ . rw [h₇₀, mul_one] at h₂ rw [h₂] at h₃ linarith [h₃] . rw [h₇₁] at hc simp at hc lemma imo_2022_p5_8_17 (a b p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) -- (hp : Nat.Prime p) -- (h₁ : a ^ p = b ! + p) -- (hbp : p ≤ b) -- (hb2p : b < 2 * p) -- (gp : p ≤ a) -- (h₃ : p < a) (c : ℕ) -- (h₂ : a = p * c) -- (gc : 0 < c) -- (hc : c < p) -- (g₁ : c ∣ c ^ p) (h₄ : c ∣ a ^ p) (h₅ : c ∣ b !) (g₂ : p = a ^ p - b !) : c ∣ p := by rw [g₂] exact dvd_sub' h₄ h₅ lemma imo_2022_p5_8_18 (a b p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) (hp : Nat.Prime p) -- (h₁ : a ^ p = b ! + p) -- (hbp : p ≤ b) -- (hb2p : b < 2 * p) -- (gp : p ≤ a) (h₃ : p < a) (c : ℕ) (h₂ : a = p * c) (gc : 0 < c) (hc : c < p) (g₁ : c ∣ c ^ p) (h₄ : c ∣ a ^ p) (h₅ : c ∣ b !) (g₂ : p = a ^ p - b !) (h₆ : c ∣ p) : False := by have h₇: c = 1 ∨ c = p := by exact (dvd_prime hp).mp h₆ cases' h₇ with h₇₀ h₇₁ . rw [h₇₀, mul_one] at h₂ rw [h₂] at h₃ linarith [h₃] . rw [h₇₁] at hc simp at hc lemma imo_2022_p5_8_19 (a b p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) -- (hp : Nat.Prime p) -- (h₁ : a ^ p = b ! + p) -- (hbp : p ≤ b) -- (hb2p : b < 2 * p) -- (gp : p ≤ a) (h₃ : p < a) (c : ℕ) (h₂ : a = p * c) (gc : 0 < c) (hc : c < p) (g₁ : c ∣ c ^ p) (h₄ : c ∣ a ^ p) (h₅ : c ∣ b !) (g₂ : p = a ^ p - b !) (h₆ : c ∣ p) (h₇ : c = 1 ∨ c = p) : False := by cases' h₇ with h₇₀ h₇₁ . rw [h₇₀, mul_one] at h₂ rw [h₂] at h₃ linarith [h₃] . rw [h₇₁] at hc simp at hc lemma imo_2022_p5_8_20 (a b p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) -- (hp : Nat.Prime p) -- (h₁ : a ^ p = b ! + p) -- (hbp : p ≤ b) -- (hb2p : b < 2 * p) -- (gp : p ≤ a) (h₃ : p < a) (c : ℕ) (h₂ : a = p * c) (gc : 0 < c) (hc : c < p) (g₁ : c ∣ c ^ p) (h₄ : c ∣ a ^ p) (h₅ : c ∣ b !) (g₂ : p = a ^ p - b !) (h₆ : c ∣ p) (h₇₀ : c = 1) : False := by rw [h₇₀, mul_one] at h₂ rw [h₂] at h₃ linarith [h₃] lemma imo_2022_p5_8_21 (a b p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) -- (hp : Nat.Prime p) -- (h₁ : a ^ p = b ! + p) -- (hbp : p ≤ b) -- (hb2p : b < 2 * p) -- (gp : p ≤ a) -- (h₃ : p < a) (c : ℕ) -- (h₂ : a = p * c) -- (gc : 0 < c) (hc : c < p) (g₁ : c ∣ c ^ p) (h₄ : c ∣ a ^ p) (h₅ : c ∣ b !) (g₂ : p = a ^ p - b !) (h₆ : c ∣ p) (h₇₁ : c = p) : False := by rw [h₇₁] at hc simp at hc lemma imo_2022_p5_8_22 (a b p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) (hp : Nat.Prime p) (h₁ : a ^ p = b ! + p) -- (hbp : p ≤ b) (hb2p : b < 2 * p) -- (gp : p ≤ a) -- (h₃ : p < a) (c : ℕ) (h₂ : a = p * c) -- (gc : 0 < c) (hc : p ≤ c) : False := by have g₃: p^2 ≤ a := by rw [h₂, pow_two] exact mul_le_mul_left' hc p have h₃: p^(2*p) ≤ a^p := by rw [pow_mul] exact pow_left_mono p g₃ have h₇: b.factorial + p < p^(2*p) := by exact imo_2022_p5_7 b p hp hb2p rw [←h₁] at h₇ linarith lemma imo_2022_p5_8_23 (a p : ℕ) -- (b : ℕ) -- (h₀ : 0 < a ∧ 0 < b) -- (hp : Nat.Prime p) -- (h₁ : a ^ p = b ! + p) -- (hbp : p ≤ b) -- (hb2p : b < 2 * p) -- (gp : p ≤ a) -- (h₃ : p < a) (c : ℕ) (h₂ : a = p * c) -- (gc : 0 < c) (hc : p ≤ c) : p ^ 2 ≤ a := by rw [h₂, pow_two] exact mul_le_mul_left' hc p lemma imo_2022_p5_8_24 (a b p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) (hp : Nat.Prime p) (h₁ : a ^ p = b ! + p) -- (hbp : p ≤ b) (hb2p : b < 2 * p) -- (gp : p ≤ a) -- (h₃ : p < a) -- (c : ℕ) -- (h₂ : a = p * c) -- (gc : 0 < c) -- (hc : p ≤ c) (g₃ : p ^ 2 ≤ a) : False := by have h₃: p^(2*p) ≤ a^p := by rw [pow_mul] exact pow_left_mono p g₃ have h₇: b.factorial + p < p^(2*p) := by exact imo_2022_p5_7 b p hp hb2p rw [←h₁] at h₇ linarith lemma imo_2022_p5_8_25 (a p : ℕ) -- (b : ℕ) -- (h₀ : 0 < a ∧ 0 < b) -- (hp : Nat.Prime p) -- (h₁ : a ^ p = b ! + p) -- (hbp : p ≤ b) -- (hb2p : b < 2 * p) -- (gp : p ≤ a) -- (h₃ : p < a) -- (c : ℕ) -- (h₂ : a = p * c) -- (gc : 0 < c) -- (hc : p ≤ c) (g₃ : p ^ 2 ≤ a) : p ^ (2 * p) ≤ a ^ p := by rw [pow_mul] exact pow_left_mono p g₃ lemma imo_2022_p5_8_26 (a b p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) (hp : Nat.Prime p) (h₁ : a ^ p = b ! + p) -- (hbp : p ≤ b) (hb2p : b < 2 * p) -- (gp : p ≤ a) -- (h31 : p < a) -- (c : ℕ) -- (h₂ : a = p * c) -- (gc : 0 < c) -- (hc : p ≤ c) -- (g₃ : p ^ 2 ≤ a) (h₃ : p ^ (2 * p) ≤ a ^ p) : False := by have h₇: b.factorial + p < p^(2*p) := by exact imo_2022_p5_7 b p hp hb2p rw [←h₁] at h₇ linarith lemma imo_2022_p5_8_27 (a b p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) -- (hp : Nat.Prime p) (h₁ : a ^ p = b ! + p) -- (hbp : p ≤ b) -- (hb2p : b < 2 * p) -- (gp : p ≤ a) -- (h31 : p < a) -- (c : ℕ) -- (h₂ : a = p * c) -- (gc : 0 < c) -- (hc : p ≤ c) -- (g₃ : p ^ 2 ≤ a) (h₃ : p ^ (2 * p) ≤ a ^ p) (h₇ : b ! + p < p ^ (2 * p)) : False := by rw [←h₁] at h₇ linarith lemma imo_2022_p5_9 (p: ℕ) -- (hp: Nat.Prime p) (hp5: 5 ≤ p) : ((↑p:ℤ) ^ p ≡ ↑p * (↑p + 1) * (-1) ^ (p - 1) + (-1) ^ p [ZMOD (↑p + 1) ^ 2]) := by -- have h₁: ↑p ^ p = Finset.range -- binomial expansion -- take the first two elements out -- show that all the other elements are individually divisible by (p+1)^2 -- conclude that their sum is divisible by (p+1)^2 -- summation ≡ 0 [ZMOD (↑p + 1) ^ 2] -- now show that Nat.modeq.add have h₀: (↑p:ℤ) = (↑p + 1) - 1 := by simp have h₁: ↑p ^ p ≡ ((↑p + 1) - 1) ^ p [ZMOD (↑p + 1) ^ 2] := by rw [← h₀] have h₂: (((↑p:ℤ) + 1) - 1) ^ p = (↑p * (↑p + 1) * (-1:ℤ) ^ (p - 1) + (-1) ^ p) + (Finset.Ico 2 (p + 1)).sum (fun (k : ℕ) => (↑p + 1) ^ k * (-1:ℤ) ^ (p - k) * ↑(p.choose k)) := by rw [sub_eq_add_neg] rw [add_pow ((↑p:ℤ) + 1) (-1:ℤ)] have g₀: 2 ≤ p + 1 := by have gg₀: 5 + 1 ≤ p + 1 := by exact add_le_add_right hp5 1 refine le_trans ?_ gg₀ norm_num have g₁: 1 ≤ 2 := by norm_num rw [← Finset.sum_range_add_sum_Ico _ g₀] rw [← Finset.sum_range_add_sum_Ico _ g₁] simp rw [add_comm] simp rw [mul_comm] rw [mul_assoc] have h₃: 0 ≡ (Finset.Ico 2 (p + 1)).sum (fun (k : ℕ) => (↑p + 1) ^ k * (-1) ^ (p - k) * ↑(p.choose k)) [ZMOD (↑p + 1) ^ 2] := by refine Int.modEq_of_dvd ?_ simp refine Finset.dvd_sum ?_ intros x g₀ have gx: 2 ≤ x := by exact (Finset.mem_Ico.mp g₀).left rw [mul_assoc] refine dvd_mul_of_dvd_left ?_ ((-1:ℤ) ^ (p - x) * ↑(p.choose x)) refine pow_dvd_pow ((↑p:ℤ) + 1) gx rw [h₂] at h₁ rw [← add_zero ((↑p:ℤ) ^ p)] at h₁ exact Int.ModEq.add_right_cancel h₃ h₁ lemma imo_2022_p5_9_1 (p : ℕ) (hp5 : 5 ≤ p) -- (h₀ : ↑p = ↑p + 1 - 1) (h₁ : ↑p ^ p ≡ (↑p + 1 - 1) ^ p [ZMOD (↑p + 1) ^ 2]) : ↑p ^ p ≡ ↑p * (↑p + 1) * (-1) ^ (p - 1) + (-1) ^ p [ZMOD (↑p + 1) ^ 2] := by have h₂: (((↑p:ℤ) + 1) - 1) ^ p = (↑p * (↑p + 1) * (-1:ℤ) ^ (p - 1) + (-1) ^ p) + (Finset.Ico 2 (p + 1)).sum (fun (k : ℕ) => (↑p + 1) ^ k * (-1:ℤ) ^ (p - k) * ↑(p.choose k)) := by rw [sub_eq_add_neg] rw [add_pow ((↑p:ℤ) + 1) (-1:ℤ)] have g₀: 2 ≤ p + 1 := by have gg₀: 5 + 1 ≤ p + 1 := by exact add_le_add_right hp5 1 refine le_trans ?_ gg₀ norm_num have g₁: 1 ≤ 2 := by norm_num rw [← Finset.sum_range_add_sum_Ico _ g₀] rw [← Finset.sum_range_add_sum_Ico _ g₁] simp rw [add_comm] simp rw [mul_comm] rw [mul_assoc] have h₃: 0 ≡ (Finset.Ico 2 (p + 1)).sum (fun (k : ℕ) => (↑p + 1) ^ k * (-1) ^ (p - k) * ↑(p.choose k)) [ZMOD (↑p + 1) ^ 2] := by refine Int.modEq_of_dvd ?_ simp refine Finset.dvd_sum ?_ intros x g₀ have gx: 2 ≤ x := by exact (Finset.mem_Ico.mp g₀).left rw [mul_assoc] refine dvd_mul_of_dvd_left ?_ ((-1:ℤ) ^ (p - x) * ↑(p.choose x)) refine pow_dvd_pow ((↑p:ℤ) + 1) gx rw [h₂] at h₁ rw [← add_zero ((↑p:ℤ) ^ p)] at h₁ exact Int.ModEq.add_right_cancel h₃ h₁ lemma imo_2022_p5_9_2 (p : ℕ) (hp5 : 5 ≤ p) : -- (h₀ : ↑p = ↑p + 1 - 1) -- (h₁ : ↑p ^ p ≡ (↑p + 1 - 1) ^ p [ZMOD (↑p + 1) ^ 2]) : (↑p + 1 - 1) ^ p = ↑p * (↑p + 1) * (-1:ℤ) ^ (p - 1) + (-1) ^ p + Finset.sum (Finset.Ico 2 (p + 1)) fun k => (↑p + 1) ^ k * (-1:ℤ) ^ (p - k) * ↑(choose p k) := by rw [sub_eq_add_neg] rw [add_pow ((↑p:ℤ) + 1) (-1:ℤ)] have g₀: 2 ≤ p + 1 := by have gg₀: 5 + 1 ≤ p + 1 := by exact add_le_add_right hp5 1 refine le_trans ?_ gg₀ norm_num have g₁: 1 ≤ 2 := by norm_num rw [← Finset.sum_range_add_sum_Ico _ g₀] rw [← Finset.sum_range_add_sum_Ico _ g₁] simp rw [add_comm] simp rw [mul_comm] rw [mul_assoc] lemma imo_2022_p5_9_3 (p : ℕ) (hp5 : 5 ≤ p) : -- (h₀ : ↑p = ↑p + 1 - 1) -- (h₁ : ↑p ^ p ≡ (↑p + 1 - 1) ^ p [ZMOD (↑p + 1) ^ 2]) (Finset.sum (Finset.range (p + 1)) fun m => ((↑p:ℤ) + 1) ^ m * (-1:ℤ) ^ (p - m) * ↑(choose p m)) = (↑p:ℤ) * ((↑p:ℤ) + 1) * (-1:ℤ) ^ (p - 1) + (-1) ^ p + Finset.sum (Finset.Ico 2 (p + 1)) fun k => ((↑p:ℤ) + 1) ^ k * (-1:ℤ) ^ (p - k) * ↑(choose p k) := by have g₀: 2 ≤ p + 1 := by have gg₀: 5 + 1 ≤ p + 1 := by exact add_le_add_right hp5 1 refine le_trans ?_ gg₀ norm_num have g₁: 1 ≤ 2 := by norm_num rw [← Finset.sum_range_add_sum_Ico _ g₀] rw [← Finset.sum_range_add_sum_Ico _ g₁] simp rw [add_comm] simp rw [mul_comm] rw [mul_assoc] lemma imo_2022_p5_9_4 (p : ℕ) (hp5 : 5 ≤ p) : -- (h₀ : ↑p = ↑p + 1 - 1) -- (h₁ : ↑p ^ p ≡ (↑p + 1 - 1) ^ p [ZMOD (↑p + 1) ^ 2]) : (Finset.sum (Finset.range (p + 1)) fun m => ((↑p + 1) ^ m * (-1:ℤ) ^ (p - m) * ↑(choose p m))) = ↑p * (↑p + 1) * (-1:ℤ) ^ (p - 1) + (-1:ℤ) ^ p + Finset.sum (Finset.Ico 2 (p + 1)) fun k => (↑p + 1) ^ k * (-1:ℤ) ^ (p - k) * ↑(choose p k) := by have g₀: 2 ≤ p + 1 := by have gg₀: 5 + 1 ≤ p + 1 := by exact add_le_add_right hp5 1 refine le_trans ?_ gg₀ norm_num have g₁: 1 ≤ 2 := by norm_num rw [← Finset.sum_range_add_sum_Ico _ g₀] rw [← Finset.sum_range_add_sum_Ico _ g₁] simp rw [add_comm] simp rw [mul_comm] rw [mul_assoc] lemma imo_2022_p5_9_5 (p : ℕ) (hp5 : 5 ≤ p) : -- (h₀ : ↑p = ↑p + 1 - 1) -- (h₁ : ↑p ^ p ≡ (↑p + 1 - 1) ^ p [ZMOD (↑p + 1) ^ 2]) : 2 ≤ p + 1 := by have gg₀: 5 + 1 ≤ p + 1 := by exact add_le_add_right hp5 1 refine le_trans ?_ gg₀ norm_num lemma imo_2022_p5_9_6 (p : ℕ) -- (hp5 : 5 ≤ p) -- (h₀ : ↑p = ↑p + 1 - 1) -- (h₁ : ↑p ^ p ≡ (↑p + 1 - 1) ^ p [ZMOD (↑p + 1) ^ 2]) (g₀ : 2 ≤ p + 1) : (Finset.sum (Finset.range (p + 1)) fun m => (↑p + 1) ^ m * (-1:ℤ) ^ (p - m) * ↑(choose p m)) = ↑p * (↑p + 1) * (-1) ^ (p - 1) + (-1) ^ p + Finset.sum (Finset.Ico 2 (p + 1)) fun k => (↑p + 1) ^ k * (-1:ℤ) ^ (p - k) * ↑(choose p k) := by have g₁: 1 ≤ 2 := by norm_num rw [← Finset.sum_range_add_sum_Ico _ g₀] rw [← Finset.sum_range_add_sum_Ico _ g₁] simp rw [add_comm] simp rw [mul_comm] rw [mul_assoc] lemma imo_2022_p5_9_7 (p : ℕ) : -- (hp5 : 5 ≤ p) -- (h₀ : ↑p = ↑p + 1 - 1) -- (h₁ : ↑p ^ p ≡ (↑p + 1 - 1) ^ p [ZMOD (↑p + 1) ^ 2]) -- (g₀ : 2 ≤ p + 1) -- (g₁ : 1 ≤ 2) : (((Finset.sum (Finset.range 1) fun k => (↑p + 1) ^ k * (-1:ℤ) ^ (p - k) * ↑(choose p k)) + Finset.sum (Finset.Ico 1 2) fun k => (↑p + 1) ^ k * (-1:ℤ) ^ (p - k) * ↑(choose p k)) + Finset.sum (Finset.Ico 2 (p + 1)) fun k => (↑p + 1) ^ k * (-1:ℤ) ^ (p - k) * ↑(choose p k)) = ↑p * (↑p + 1) * (-1) ^ (p - 1) + (-1) ^ p + Finset.sum (Finset.Ico 2 (p + 1)) fun k => (↑p + 1) ^ k * (-1:ℤ) ^ (p - k) * ↑(choose p k) := by simp rw [add_comm] simp rw [mul_comm] rw [mul_assoc] lemma imo_2022_p5_9_8 (p : ℕ) : -- (hp5 : 5 ≤ p) -- (h₀ : ↑p = ↑p + 1 - 1) -- (h₁ : ↑p ^ p ≡ (↑p + 1 - 1) ^ p [ZMOD (↑p + 1) ^ 2]) -- (g₀ : 2 ≤ p + 1) -- (g₁ : 1 ≤ 2) : (-1:ℤ) ^ p + (↑p + 1) * (-1) ^ (p - 1) * ↑p = ↑p * (↑p + 1) * (-1) ^ (p - 1) + (-1) ^ p := by rw [add_comm] simp rw [mul_comm] rw [mul_assoc] lemma imo_2022_p5_9_9 (p : ℕ) : -- (hp5 : 5 ≤ p) -- (h₀ : ↑p = ↑p + 1 - 1) -- (h₁ : ↑p ^ p ≡ (↑p + 1 - 1) ^ p [ZMOD (↑p + 1) ^ 2]) -- (g₀ : 2 ≤ p + 1) -- (g₁ : 1 ≤ 2) : (↑p + 1) * (-1:ℤ) ^ (p - 1) * ↑p = ↑p * (↑p + 1) * (-1) ^ (p - 1) := by rw [mul_comm] rw [mul_assoc] lemma imo_2022_p5_9_10 (p : ℕ) (h₀: (↑p + 1) * (-1:ℤ) ^ (p - 1) * ↑p + (-1) ^ p = ↑p * (↑p + 1) * (-1) ^ (p - 1) + (-1) ^ p) -- (hp5 : 5 ≤ p) -- (h₀ : ↑p = ↑p + 1 - 1) -- (h₁ : ↑p ^ p ≡ (↑p + 1 - 1) ^ p [ZMOD (↑p + 1) ^ 2]) (g₀ : 2 ≤ p + 1) : (Finset.sum (Finset.range (p + 1)) fun m => (↑p + 1) ^ m * (-1:ℤ) ^ (p - m) * ↑(choose p m)) = ↑p * (↑p + 1) * (-1) ^ (p - 1) + (-1) ^ p + Finset.sum (Finset.Ico 2 (p + 1)) fun k => (↑p + 1) ^ k * (-1:ℤ) ^ (p - k) * ↑(choose p k) := by have g₁: 1 ≤ 2 := by norm_num rw [← Finset.sum_range_add_sum_Ico _ g₀] rw [← Finset.sum_range_add_sum_Ico _ g₁] simp rw [add_comm] exact h₀ lemma imo_2022_p5_9_11 (p : ℕ) -- (hp5 : 5 ≤ p) -- (h₀ : ↑p = ↑p + 1 - 1) (h₁ : ↑p ^ p ≡ (↑p + 1 - 1) ^ p [ZMOD (↑p + 1) ^ 2]) (h₂ : (↑p + 1 - 1) ^ p = ↑p * (↑p + 1) * (-1:ℤ) ^ (p - 1) + (-1) ^ p + Finset.sum (Finset.Ico 2 (p + 1)) fun k => (↑p + 1) ^ k * (-1:ℤ) ^ (p - k) * ↑(choose p k)) : ↑p ^ p ≡ ↑p * (↑p + 1) * (-1) ^ (p - 1) + (-1) ^ p [ZMOD (↑p + 1) ^ 2] := by have h₃: 0 ≡ (Finset.Ico 2 (p + 1)).sum (fun (k : ℕ) => (↑p + 1) ^ k * (-1) ^ (p - k) * ↑(p.choose k)) [ZMOD (↑p + 1) ^ 2] := by refine Int.modEq_of_dvd ?_ simp refine Finset.dvd_sum ?_ intros x g₀ have gx: 2 ≤ x := by exact (Finset.mem_Ico.mp g₀).left rw [mul_assoc] refine dvd_mul_of_dvd_left ?_ ((-1:ℤ) ^ (p - x) * ↑(p.choose x)) refine pow_dvd_pow ((↑p:ℤ) + 1) gx rw [h₂] at h₁ rw [← add_zero ((↑p:ℤ) ^ p)] at h₁ exact Int.ModEq.add_right_cancel h₃ h₁ lemma imo_2022_p5_9_12 (p : ℕ) : -- (hp5 : 5 ≤ p) -- (h₀ : ↑p = ↑p + 1 - 1) -- (h₁ : ↑p ^ p ≡ (↑p + 1 - 1) ^ p [ZMOD (↑p + 1) ^ 2]) -- (h₂ : (↑p + 1 - 1) ^ p = -- ↑p * (↑p + 1) * (-1:ℤ) ^ (p - 1) + (-1) ^ p + -- Finset.sum (Finset.Ico 2 (p + 1)) fun k => (↑p + 1) ^ k * (-1:ℤ) ^ (p - k) * ↑(choose p k)) : 0 ≡ Finset.sum (Finset.Ico 2 (p + 1)) fun k => (↑p + 1) ^ k * (-1) ^ (p - k) * ↑(choose p k) [ZMOD (↑p + 1) ^ 2] := by refine Int.modEq_of_dvd ?_ simp refine Finset.dvd_sum ?_ intros x g₀ have gx: 2 ≤ x := by exact (Finset.mem_Ico.mp g₀).left rw [mul_assoc] refine dvd_mul_of_dvd_left ?_ ((-1:ℤ) ^ (p - x) * ↑(p.choose x)) refine pow_dvd_pow ((↑p:ℤ) + 1) gx lemma imo_2022_p5_9_13 (p : ℕ) : -- (hp5 : 5 ≤ p) -- (h₀ : ↑p = ↑p + 1 - 1) -- (h₁ : ↑p ^ p ≡ (↑p + 1 - 1) ^ p [ZMOD (↑p + 1) ^ 2]) -- (h₂ : (↑p + 1 - 1) ^ p = -- ↑p * (↑p + 1) * (-1:ℤ) ^ (p - 1) + (-1) ^ p + -- Finset.sum (Finset.Ico 2 (p + 1)) fun k => (↑p + 1) ^ k * (-1:ℤ) ^ (p - k) * ↑(choose p k)) -- (h₃: 0 ≡ Finset.sum (Finset.Ico 2 (p + 1)) -- fun (k:ℕ) => (↑p + 1) ^ k * (-1) ^ (p - k) * ↑(choose p k) [ZMOD (↑p + 1) ^ 2]) : ((↑p:ℤ) + 1) ^ 2 ∣ Finset.sum (Finset.Ico 2 (p + 1)) fun (k:ℕ) => ((↑p:ℤ) + 1) ^ k * (-1:ℤ) ^ (p - k) * ↑(choose p k) := by refine Finset.dvd_sum ?_ intros x g₀ have gx: 2 ≤ x := by exact (Finset.mem_Ico.mp g₀).left rw [mul_assoc] refine dvd_mul_of_dvd_left ?_ ((-1:ℤ) ^ (p - x) * ↑(p.choose x)) exact pow_dvd_pow ((↑p:ℤ) + 1) gx lemma imo_2022_p5_9_14 (p : ℕ) -- (hp5 : 5 ≤ p) -- (h₀ : ↑p = ↑p + 1 - 1) -- (h₁ : ↑p ^ p ≡ (↑p + 1 - 1) ^ p [ZMOD (↑p + 1) ^ 2]) -- (h₂ : (↑p + 1 - 1) ^ p = -- ↑p * (↑p + 1) * (-1:ℤ) ^ (p - 1) + (-1) ^ p + -- Finset.sum (Finset.Ico 2 (p + 1)) fun k => (↑p + 1) ^ k * (-1:ℤ) ^ (p - k) * ↑(choose p k)) (h₃ : ∀ i ∈ Finset.Ico 2 (p + 1), ((↑p:ℤ) + 1) ^ 2 ∣ (↑p + 1) ^ i * (-1:ℤ) ^ (p - i) * ↑(choose p i)) : 0 ≡ Finset.sum (Finset.Ico 2 (p + 1)) fun k => (↑p + 1) ^ k * (-1) ^ (p - k) * ↑(choose p k) [ZMOD (↑p + 1) ^ 2] := by refine Int.modEq_of_dvd ?_ simp exact Finset.dvd_sum h₃ lemma imo_2022_p5_9_15 (p : ℕ) -- (hp5 : 5 ≤ p) -- (h₀ : ↑p = ↑p + 1 - 1) -- (h₁ : ↑p ^ p ≡ (↑p + 1 - 1) ^ p [ZMOD (↑p + 1) ^ 2]) (h₂ : ∀ x ∈ Finset.Ico 2 (p + 1), ((↑p:ℤ) + 1) ^ 2 ∣ ((↑p:ℤ) + 1) ^ x) : ((↑p:ℤ) + 1) ^ 2 ∣ Finset.sum (Finset.Ico 2 (p + 1)) fun k => (↑p + 1) ^ k * (-1:ℤ) ^ (p - k) * ↑(choose p k) := by refine Finset.dvd_sum ?_ intros x g₀ rw [mul_assoc] refine dvd_mul_of_dvd_left ?_ ((-1:ℤ) ^ (p - x) * ↑(p.choose x)) exact h₂ x g₀ lemma imo_2022_p5_9_16 (p : ℕ) : -- (hp5 : 5 ≤ p) -- (h₀ : ↑p = ↑p + 1 - 1) -- (h₁ : ↑p ^ p ≡ (↑p + 1 - 1) ^ p [ZMOD (↑p + 1) ^ 2]) -- (h₂ : (↑p + 1 - 1) ^ p = -- ↑p * (↑p + 1) * (-1:ℤ) ^ (p - 1) + (-1) ^ p + -- Finset.sum (Finset.Ico 2 (p + 1)) fun k => (↑p + 1) ^ k * (-1:ℤ) ^ (p - k) * ↑(choose p k)) : ∀ i ∈ Finset.Ico 2 (p + 1), ((↑p:ℤ) + 1) ^ 2 ∣ (↑p + 1) ^ i * (-1:ℤ) ^ (p - i) * ↑(choose p i) := by intros x g₀ have gx: 2 ≤ x := by exact (Finset.mem_Ico.mp g₀).left rw [mul_assoc] refine dvd_mul_of_dvd_left ?_ ((-1:ℤ) ^ (p - x) * ↑(p.choose x)) refine pow_dvd_pow ((↑p:ℤ) + 1) gx lemma imo_2022_p5_9_17 (p : ℕ) -- (hp5 : 5 ≤ p) -- (h₀ : ↑p = ↑p + 1 - 1) -- (h₁ : ↑p ^ p ≡ (↑p + 1 - 1) ^ p [ZMOD (↑p + 1) ^ 2]) -- (h₂ : (↑p + 1 - 1) ^ p = -- ↑p * (↑p + 1) * (-1:ℤ) ^ (p - 1) + (-1) ^ p + -- Finset.sum (Finset.Ico 2 (p + 1)) fun k => (↑p + 1) ^ k * (-1:ℤ) ^ (p - k) * ↑(choose p k)) (x : ℕ) -- (g₀ : x ∈ Finset.Ico 2 (p + 1)) (gx : 2 ≤ x) : ((↑p:ℤ) + 1) ^ 2 ∣ (↑p + 1) ^ x * (-1:ℤ) ^ (p - x) * ↑(choose p x) := by rw [mul_assoc] refine dvd_mul_of_dvd_left ?_ ((-1:ℤ) ^ (p - x) * ↑(p.choose x)) refine pow_dvd_pow ((↑p:ℤ) + 1) gx lemma imo_2022_p5_9_18 (p : ℕ) -- (hp5 : 5 ≤ p) -- (h₀ : ↑p = ↑p + 1 - 1) -- (h₁ : ↑p ^ p ≡ (↑p + 1 - 1) ^ p [ZMOD (↑p + 1) ^ 2]) -- (h₂ : (↑p + 1 - 1) ^ p = -- ↑p * (↑p + 1) * (-1:ℤ) ^ (p - 1) + (-1) ^ p + -- Finset.sum (Finset.Ico 2 (p + 1)) fun k => (↑p + 1) ^ k * (-1:ℤ) ^ (p - k) * ↑(choose p k)) (x : ℕ) (g₀ : x ∈ Finset.Ico 2 (p + 1)) : ((↑p:ℤ) + 1) ^ 2 ∣ ((↑p:ℤ) + 1) ^ x := by refine pow_dvd_pow ((↑p:ℤ) + 1) ?_ exact (Finset.mem_Ico.mp g₀).left lemma imo_2022_p5_9_19 (p : ℕ) -- (hp5 : 5 ≤ p) -- (h₀ : ↑p = ↑p + 1 - 1) (h₁ : ↑p ^ p ≡ (↑p + 1 - 1) ^ p [ZMOD (↑p + 1) ^ 2]) (h₂ : (↑p + 1 - 1) ^ p = ↑p * (↑p + 1) * (-1:ℤ) ^ (p - 1) + (-1) ^ p + Finset.sum (Finset.Ico 2 (p + 1)) fun k => (↑p + 1) ^ k * (-1:ℤ) ^ (p - k) * ↑(choose p k)) (h₃ : 0 ≡ Finset.sum (Finset.Ico 2 (p + 1)) fun k => (↑p + 1) ^ k * (-1) ^ (p - k) * ↑(choose p k) [ZMOD (↑p + 1) ^ 2]) : ↑p ^ p ≡ ↑p * (↑p + 1) * (-1) ^ (p - 1) + (-1) ^ p [ZMOD (↑p + 1) ^ 2] := by rw [h₂] at h₁ rw [← add_zero ((↑p:ℤ) ^ p)] at h₁ exact Int.ModEq.add_right_cancel h₃ h₁ lemma imo_2022_p5_10 (p: ℕ) (hp: Nat.Prime p) (hp5: 5 ≤ p) -- (hp7: 7 ≤ p) (h₀: (p + 1) ^ 2 ∣ p ^ p - p) : False := by have h₁: ((↑p^p - ↑p):ℤ) ≡ (↑(p.choose 1) * ↑(p + 1) * (-1:ℤ)^(p-1) + (-1:ℤ)^p) - ↑p [ZMOD ↑(p+1)^2] := by refine Int.ModEq.sub_right (↑p) ?_ simp exact imo_2022_p5_9 p hp5 have gpo: Odd p := by refine Nat.Prime.odd_of_ne_two hp ?_ linarith [hp5] have gpe: Even (p - 1) := by refine hp.even_sub_one ?_ linarith [hp5] have g₁: (-1:ℤ) ^ (p - 1) = 1 := by exact Even.neg_one_pow gpe have g₂: (-1:ℤ) ^ (p) = -1 := by exact Odd.neg_one_pow gpo rw [g₁,g₂] at h₁ simp at h₁ -- norm_cast at h₁ have h₂: (p ^ p - p) ≡ (p * (p + 1)) - 1 - p [MOD ((p + 1) ^ 2)] := by refine Int.natCast_modEq_iff.mp ?_ have g₃: p ≤ p^p := by refine Nat.le_self_pow (by linarith) _ rw [Nat.cast_sub g₃] have g₄: p ≤ p * (p + 1) - 1 := by rw [mul_add] simp rw [add_comm, Nat.add_sub_assoc] simp rw [← pow_two] refine Nat.one_le_pow 2 p (by linarith) rw [Nat.cast_sub g₄] have g₅: 1 ≤ p * (p + 1) := by rw [← mul_one (p * (p + 1))] refine Nat.le_mul_of_pos_left ?_ ?_ refine Nat.mul_pos (by linarith) (by linarith) rw [Nat.cast_sub g₅] rw [← sub_eq_add_neg] at h₁ norm_cast norm_cast at h₁ have h₃: p * (p + 1) - 1 - p = p^2 - 1 := by rw [Nat.sub_sub, mul_add] simp rw [← pow_two] exact Nat.add_sub_add_right (p^2) p 1 rw [h₃] at h₂ clear h₃ gpo gpe g₁ g₂ -- now derive a line of contradictions from h₀ have hc₁: (p ^ p - p) ≡ 0 [MOD (p+1)^2] := by exact modEq_zero_iff_dvd.mpr h₀ -- mix the contradiction with what we had in h₂ have h₄: p ^ 2 - 1 ≡ 0 [MOD (p+1)^2] := by apply Nat.ModEq.symm at h₂ exact Nat.ModEq.trans h₂ hc₁ have h₅: p - 1 ≡ 0 [MOD (p+1)] := by rw [pow_two] at h₄ have g₀: p^2 - 1^2 = (p-1) * (p+1) := by rw [mul_comm] exact Nat.sq_sub_sq p 1 simp at g₀ rw [g₀] at h₄ have g₁: p + 1 ≠ 0 := by linarith refine Nat.ModEq.mul_right_cancel' g₁ ?_ rw [zero_mul] exact h₄ have h₆: p - 1 ≤ 0 := by refine Nat.ModEq.le_of_lt_add h₅ ?_ simp rw [← succ_eq_add_one] refine Nat.sub_lt_succ p 1 have h₇: 0 < p - 1 := by simp linarith linarith [h₆,h₇] lemma imo_2022_p5_10_1 (p : ℕ) -- (hp : Nat.Prime p) (hp5 : 5 ≤ p) : -- (h₀ : (p + 1) ^ 2 ∣ p ^ p - p) : ↑p ^ p - ↑p ≡ ↑(choose p 1) * ↑(p + 1) * (-1) ^ (p - 1) + (-1) ^ p - ↑p [ZMOD ↑(p + 1) ^ 2] := by refine Int.ModEq.sub_right (↑p) ?_ simp exact imo_2022_p5_9 p hp5 lemma imo_2022_p5_10_2 (p : ℕ) (hp : Nat.Prime p) (hp5 : 5 ≤ p) (h₀ : (p + 1) ^ 2 ∣ p ^ p - p) (h₁ : ↑p ^ p - ↑p ≡ ↑(choose p 1) * ↑(p + 1) * (-1) ^ (p - 1) + (-1) ^ p - ↑p [ZMOD ↑(p + 1) ^ 2]) : False := by have gpo: Odd p := by refine Nat.Prime.odd_of_ne_two hp ?_ linarith [hp5] have gpe: Even (p - 1) := by refine hp.even_sub_one ?_ linarith [hp5] have g₁: (-1:ℤ) ^ (p - 1) = 1 := by exact Even.neg_one_pow gpe have g₂: (-1:ℤ) ^ (p) = -1 := by exact Odd.neg_one_pow gpo rw [g₁,g₂] at h₁ simp at h₁ -- norm_cast at h₁ have h₂: (p ^ p - p) ≡ (p * (p + 1)) - 1 - p [MOD ((p + 1) ^ 2)] := by refine Int.natCast_modEq_iff.mp ?_ have g₃: p ≤ p^p := by refine Nat.le_self_pow (by linarith) _ rw [Nat.cast_sub g₃] have g₄: p ≤ p * (p + 1) - 1 := by rw [mul_add] simp rw [add_comm, Nat.add_sub_assoc] simp rw [← pow_two] refine Nat.one_le_pow 2 p (by linarith) rw [Nat.cast_sub g₄] have g₅: 1 ≤ p * (p + 1) := by rw [← mul_one (p * (p + 1))] refine Nat.le_mul_of_pos_left ?_ ?_ refine Nat.mul_pos (by linarith) (by linarith) rw [Nat.cast_sub g₅] rw [← sub_eq_add_neg] at h₁ norm_cast norm_cast at h₁ have h₃: p * (p + 1) - 1 - p = p^2 - 1 := by rw [Nat.sub_sub, mul_add] simp rw [← pow_two] exact Nat.add_sub_add_right (p^2) p 1 rw [h₃] at h₂ clear h₃ gpo gpe g₁ g₂ -- now derive a line of contradictions from h₀ have hc₁: (p ^ p - p) ≡ 0 [MOD (p+1)^2] := by exact modEq_zero_iff_dvd.mpr h₀ -- mix the contradiction with what we had in h₂ have h₄: p ^ 2 - 1 ≡ 0 [MOD (p+1)^2] := by apply Nat.ModEq.symm at h₂ exact Nat.ModEq.trans h₂ hc₁ have h₅: p - 1 ≡ 0 [MOD (p+1)] := by rw [pow_two] at h₄ have g₀: p^2 - 1^2 = (p-1) * (p+1) := by rw [mul_comm] exact Nat.sq_sub_sq p 1 simp at g₀ rw [g₀] at h₄ have g₁: p + 1 ≠ 0 := by linarith refine Nat.ModEq.mul_right_cancel' g₁ ?_ rw [zero_mul] exact h₄ have h₆: p - 1 ≤ 0 := by refine Nat.ModEq.le_of_lt_add h₅ ?_ simp rw [← succ_eq_add_one] refine Nat.sub_lt_succ p 1 have h₇: 0 < p - 1 := by simp linarith linarith [h₆,h₇] lemma imo_2022_p5_10_3 (p : ℕ) (hp : Nat.Prime p) (hp5 : 5 ≤ p) : -- (h₀ : (p + 1) ^ 2 ∣ p ^ p - p) -- (h₁ : ↑p ^ p - ↑p ≡ ↑(choose p 1) * ↑(p + 1) * (-1) ^ (p - 1) + (-1) ^ p - ↑p [ZMOD ↑(p + 1) ^ 2]) : Odd p := by refine Nat.Prime.odd_of_ne_two hp ?_ linarith [hp5] lemma imo_2022_p5_10_4 (p : ℕ) (hp : Nat.Prime p) (hp5 : 5 ≤ p) (h₀ : (p + 1) ^ 2 ∣ p ^ p - p) (h₁ : ↑p ^ p - ↑p ≡ ↑(choose p 1) * ↑(p + 1) * (-1) ^ (p - 1) + (-1) ^ p - ↑p [ZMOD ↑(p + 1) ^ 2]) (gpo : Odd p) : False := by have gpe: Even (p - 1) := by refine hp.even_sub_one ?_ linarith [hp5] have g₁: (-1:ℤ) ^ (p - 1) = 1 := by exact Even.neg_one_pow gpe have g₂: (-1:ℤ) ^ (p) = -1 := by exact Odd.neg_one_pow gpo rw [g₁,g₂] at h₁ simp at h₁ -- norm_cast at h₁ have h₂: (p ^ p - p) ≡ (p * (p + 1)) - 1 - p [MOD ((p + 1) ^ 2)] := by refine Int.natCast_modEq_iff.mp ?_ have g₃: p ≤ p^p := by refine Nat.le_self_pow (by linarith) _ rw [Nat.cast_sub g₃] have g₄: p ≤ p * (p + 1) - 1 := by rw [mul_add] simp rw [add_comm, Nat.add_sub_assoc] simp rw [← pow_two] refine Nat.one_le_pow 2 p (by linarith) rw [Nat.cast_sub g₄] have g₅: 1 ≤ p * (p + 1) := by rw [← mul_one (p * (p + 1))] refine Nat.le_mul_of_pos_left ?_ ?_ refine Nat.mul_pos (by linarith) (by linarith) rw [Nat.cast_sub g₅] rw [← sub_eq_add_neg] at h₁ norm_cast norm_cast at h₁ have h₃: p * (p + 1) - 1 - p = p^2 - 1 := by rw [Nat.sub_sub, mul_add] simp rw [← pow_two] exact Nat.add_sub_add_right (p^2) p 1 rw [h₃] at h₂ clear h₃ gpo gpe g₁ g₂ -- now derive a line of contradictions from h₀ have hc₁: (p ^ p - p) ≡ 0 [MOD (p+1)^2] := by exact modEq_zero_iff_dvd.mpr h₀ -- mix the contradiction with what we had in h₂ have h₄: p ^ 2 - 1 ≡ 0 [MOD (p+1)^2] := by apply Nat.ModEq.symm at h₂ exact Nat.ModEq.trans h₂ hc₁ have h₅: p - 1 ≡ 0 [MOD (p+1)] := by rw [pow_two] at h₄ have g₀: p^2 - 1^2 = (p-1) * (p+1) := by rw [mul_comm] exact Nat.sq_sub_sq p 1 simp at g₀ rw [g₀] at h₄ have g₁: p + 1 ≠ 0 := by linarith refine Nat.ModEq.mul_right_cancel' g₁ ?_ rw [zero_mul] exact h₄ have h₆: p - 1 ≤ 0 := by refine Nat.ModEq.le_of_lt_add h₅ ?_ simp rw [← succ_eq_add_one] refine Nat.sub_lt_succ p 1 have h₇: 0 < p - 1 := by simp linarith linarith [h₆,h₇] lemma imo_2022_p5_10_5 (p : ℕ) (hp : Nat.Prime p) (hp5 : 5 ≤ p) : -- (h₀ : (p + 1) ^ 2 ∣ p ^ p - p) -- (h₁ : ↑p ^ p - ↑p ≡ ↑(choose p 1) * ↑(p + 1) * (-1) ^ (p - 1) + (-1) ^ p - ↑p [ZMOD ↑(p + 1) ^ 2]) -- (gpo : Odd p) : Even (p - 1) := by refine hp.even_sub_one ?_ linarith [hp5] lemma imo_2022_p5_10_6 (p : ℕ) -- (hp : Nat.Prime p) (hp5 : 5 ≤ p) (h₀ : (p + 1) ^ 2 ∣ p ^ p - p) (h₁ : ↑p ^ p - ↑p ≡ ↑(choose p 1) * ↑(p + 1) * (-1) ^ (p - 1) + (-1) ^ p - ↑p [ZMOD ↑(p + 1) ^ 2]) (gpo : Odd p) (gpe : Even (p - 1)) : False := by have g₁: (-1:ℤ) ^ (p - 1) = 1 := by exact Even.neg_one_pow gpe have g₂: (-1:ℤ) ^ (p) = -1 := by exact Odd.neg_one_pow gpo rw [g₁,g₂] at h₁ simp at h₁ have h₂: (p ^ p - p) ≡ (p * (p + 1)) - 1 - p [MOD ((p + 1) ^ 2)] := by refine Int.natCast_modEq_iff.mp ?_ have g₃: p ≤ p^p := by refine Nat.le_self_pow (by linarith) _ rw [Nat.cast_sub g₃] have g₄: p ≤ p * (p + 1) - 1 := by rw [mul_add] simp rw [add_comm, Nat.add_sub_assoc] simp rw [← pow_two] refine Nat.one_le_pow 2 p (by linarith) rw [Nat.cast_sub g₄] have g₅: 1 ≤ p * (p + 1) := by rw [← mul_one (p * (p + 1))] refine Nat.le_mul_of_pos_left ?_ ?_ refine Nat.mul_pos (by linarith) (by linarith) rw [Nat.cast_sub g₅] rw [← sub_eq_add_neg] at h₁ norm_cast norm_cast at h₁ have h₃: p * (p + 1) - 1 - p = p^2 - 1 := by rw [Nat.sub_sub, mul_add] simp rw [← pow_two] exact Nat.add_sub_add_right (p^2) p 1 rw [h₃] at h₂ clear h₃ gpo gpe g₁ g₂ -- now derive a line of contradictions from h₀ have hc₁: (p ^ p - p) ≡ 0 [MOD (p+1)^2] := by exact modEq_zero_iff_dvd.mpr h₀ -- mix the contradiction with what we had in h₂ have h₄: p ^ 2 - 1 ≡ 0 [MOD (p+1)^2] := by apply Nat.ModEq.symm at h₂ exact Nat.ModEq.trans h₂ hc₁ have h₅: p - 1 ≡ 0 [MOD (p+1)] := by rw [pow_two] at h₄ have g₀: p^2 - 1^2 = (p-1) * (p+1) := by rw [mul_comm] exact Nat.sq_sub_sq p 1 simp at g₀ rw [g₀] at h₄ have g₁: p + 1 ≠ 0 := by linarith refine Nat.ModEq.mul_right_cancel' g₁ ?_ rw [zero_mul] exact h₄ have h₆: p - 1 ≤ 0 := by refine Nat.ModEq.le_of_lt_add h₅ ?_ simp rw [← succ_eq_add_one] refine Nat.sub_lt_succ p 1 have h₇: 0 < p - 1 := by simp linarith linarith [h₆,h₇] lemma imo_2022_p5_10_7 (p : ℕ) -- (hp : Nat.Prime p) (hp5 : 5 ≤ p) (h₀ : (p + 1) ^ 2 ∣ p ^ p - p) (h₁ : ↑p ^ p - ↑p ≡ ↑(choose p 1) * ↑(p + 1) * (-1) ^ (p - 1) + (-1) ^ p - ↑p [ZMOD ↑(p + 1) ^ 2]) (gpo : Odd p) (gpe : Even (p - 1)) (g₁ : (-1) ^ (p - 1) = 1) (g₂ : (-1) ^ p = -1) : False := by rw [g₁,g₂] at h₁ simp at h₁ have h₂: (p ^ p - p) ≡ (p * (p + 1)) - 1 - p [MOD ((p + 1) ^ 2)] := by refine Int.natCast_modEq_iff.mp ?_ have g₃: p ≤ p^p := by refine Nat.le_self_pow (by linarith) _ rw [Nat.cast_sub g₃] have g₄: p ≤ p * (p + 1) - 1 := by rw [mul_add] simp rw [add_comm, Nat.add_sub_assoc] simp rw [← pow_two] refine Nat.one_le_pow 2 p (by linarith) rw [Nat.cast_sub g₄] have g₅: 1 ≤ p * (p + 1) := by rw [← mul_one (p * (p + 1))] refine Nat.le_mul_of_pos_left ?_ ?_ refine Nat.mul_pos (by linarith) (by linarith) rw [Nat.cast_sub g₅] rw [← sub_eq_add_neg] at h₁ norm_cast norm_cast at h₁ have h₃: p * (p + 1) - 1 - p = p^2 - 1 := by rw [Nat.sub_sub, mul_add] simp rw [← pow_two] exact Nat.add_sub_add_right (p^2) p 1 rw [h₃] at h₂ clear h₃ gpo gpe g₁ g₂ -- now derive a line of contradictions from h₀ have hc₁: (p ^ p - p) ≡ 0 [MOD (p+1)^2] := by exact modEq_zero_iff_dvd.mpr h₀ -- mix the contradiction with what we had in h₂ have h₄: p ^ 2 - 1 ≡ 0 [MOD (p+1)^2] := by apply Nat.ModEq.symm at h₂ exact Nat.ModEq.trans h₂ hc₁ have h₅: p - 1 ≡ 0 [MOD (p+1)] := by rw [pow_two] at h₄ have g₀: p^2 - 1^2 = (p-1) * (p+1) := by rw [mul_comm] exact Nat.sq_sub_sq p 1 simp at g₀ rw [g₀] at h₄ have g₁: p + 1 ≠ 0 := by linarith refine Nat.ModEq.mul_right_cancel' g₁ ?_ rw [zero_mul] exact h₄ have h₆: p - 1 ≤ 0 := by refine Nat.ModEq.le_of_lt_add h₅ ?_ simp rw [← succ_eq_add_one] refine Nat.sub_lt_succ p 1 have h₇: 0 < p - 1 := by simp linarith linarith [h₆,h₇] lemma imo_2022_p5_10_8 (p : ℕ) -- (hp : Nat.Prime p) (hp5 : 5 ≤ p) : -- (h₀ : (p + 1) ^ 2 ∣ p ^ p - p) -- (h₁ : ↑p ^ p - ↑p ≡ ↑(choose p 1) * ↑(p + 1) * (-1) ^ (p - 1) + (-1:ℤ) ^ p - ↑p [ZMOD ↑(p + 1) ^ 2]) -- (gpo : Odd p) -- (gpe : Even (p - 1)) -- (g₁ : (-1) ^ (p - 1) = 1) -- (g₂ : (-1) ^ p = -1) : ((↑p^p - ↑p):ℤ) ≡ (↑(p.choose 1) * ↑(p + 1) * (-1:ℤ)^(p-1) + (-1:ℤ)^p) - ↑p [ZMOD ↑(p+1)^2] := by refine Int.ModEq.sub_right (↑p) ?_ simp exact imo_2022_p5_9 p hp5 lemma imo_2022_p5_10_9 (p : ℕ) -- (hp : Nat.Prime p) (hp5 : 5 ≤ p) (h₀ : (p + 1) ^ 2 ∣ p ^ p - p) (gpo : Odd p) (gpe : Even (p - 1)) (g₁ : (-1) ^ (p - 1) = 1) (g₂ : (-1) ^ p = -1) (h₁ : ↑p ^ p - ↑p ≡ ↑p * (↑p + 1) + -1 - ↑p [ZMOD (↑p + 1) ^ 2]) : False := by have h₂: (p ^ p - p) ≡ (p * (p + 1)) - 1 - p [MOD ((p + 1) ^ 2)] := by refine Int.natCast_modEq_iff.mp ?_ have g₃: p ≤ p^p := by refine Nat.le_self_pow (by linarith) _ rw [Nat.cast_sub g₃] have g₄: p ≤ p * (p + 1) - 1 := by rw [mul_add] simp rw [add_comm, Nat.add_sub_assoc] simp rw [← pow_two] refine Nat.one_le_pow 2 p (by linarith) rw [Nat.cast_sub g₄] have g₅: 1 ≤ p * (p + 1) := by rw [← mul_one (p * (p + 1))] refine Nat.le_mul_of_pos_left ?_ ?_ refine Nat.mul_pos (by linarith) (by linarith) rw [Nat.cast_sub g₅] rw [← sub_eq_add_neg] at h₁ norm_cast norm_cast at h₁ have h₃: p * (p + 1) - 1 - p = p^2 - 1 := by rw [Nat.sub_sub, mul_add] simp rw [← pow_two] exact Nat.add_sub_add_right (p^2) p 1 rw [h₃] at h₂ clear h₃ gpo gpe g₁ g₂ -- now derive a line of contradictions from h₀ have hc₁: (p ^ p - p) ≡ 0 [MOD (p+1)^2] := by exact modEq_zero_iff_dvd.mpr h₀ -- mix the contradiction with what we had in h₂ have h₄: p ^ 2 - 1 ≡ 0 [MOD (p+1)^2] := by apply Nat.ModEq.symm at h₂ exact Nat.ModEq.trans h₂ hc₁ have h₅: p - 1 ≡ 0 [MOD (p+1)] := by rw [pow_two] at h₄ have g₀: p^2 - 1^2 = (p-1) * (p+1) := by rw [mul_comm] exact Nat.sq_sub_sq p 1 simp at g₀ rw [g₀] at h₄ have g₁: p + 1 ≠ 0 := by linarith refine Nat.ModEq.mul_right_cancel' g₁ ?_ rw [zero_mul] exact h₄ have h₆: p - 1 ≤ 0 := by refine Nat.ModEq.le_of_lt_add h₅ ?_ simp rw [← succ_eq_add_one] refine Nat.sub_lt_succ p 1 have h₇: 0 < p - 1 := by simp linarith linarith [h₆,h₇] lemma imo_2022_p5_10_10 (p : ℕ) -- (hp : Nat.Prime p) (hp5 : 5 ≤ p) -- (h₀ : (p + 1) ^ 2 ∣ p ^ p - p) -- (gpo : Odd p) -- (gpe : Even (p - 1)) -- (g₁ : (-1) ^ (p - 1) = 1) -- (g₂ : (-1) ^ p = -1) (h₁ : ↑p ^ p - ↑p ≡ ↑p * (↑p + 1) + -1 - ↑p [ZMOD (↑p + 1) ^ 2]) : p ^ p - p ≡ p * (p + 1) - 1 - p [MOD (p + 1) ^ 2] := by refine Int.natCast_modEq_iff.mp ?_ have g₃: p ≤ p^p := by refine Nat.le_self_pow (by linarith) _ rw [Nat.cast_sub g₃] have g₄: p ≤ p * (p + 1) - 1 := by rw [mul_add] simp rw [add_comm, Nat.add_sub_assoc] simp rw [← pow_two] refine Nat.one_le_pow 2 p (by linarith) rw [Nat.cast_sub g₄] have g₅: 1 ≤ p * (p + 1) := by rw [← mul_one (p * (p + 1))] refine Nat.le_mul_of_pos_left ?_ ?_ refine Nat.mul_pos (by linarith) (by linarith) rw [Nat.cast_sub g₅] rw [← sub_eq_add_neg] at h₁ norm_cast norm_cast at h₁ lemma imo_2022_p5_10_11 (p : ℕ) -- (hp : Nat.Prime p) (hp5 : 5 ≤ p) -- (h₀ : (p + 1) ^ 2 ∣ p ^ p - p) -- (gpo : Odd p) -- (gpe : Even (p - 1)) -- (g₁ : (-1) ^ (p - 1) = 1) -- (g₂ : (-1) ^ p = -1) (h₁ : ↑p ^ p - ↑p ≡ ↑p * (↑p + 1) + -1 - ↑p [ZMOD (↑p + 1) ^ 2]) : ↑(p ^ p - p) ≡ ↑(p * (p + 1) - 1 - p) [ZMOD ↑(((↑p:ℤ) + 1) ^ 2)] := by have g₃: p ≤ p^p := by refine Nat.le_self_pow (by linarith) _ rw [Nat.cast_sub g₃] have g₄: p ≤ p * (p + 1) - 1 := by rw [mul_add] simp rw [add_comm, Nat.add_sub_assoc] simp rw [← pow_two] refine Nat.one_le_pow 2 p (by linarith) rw [Nat.cast_sub g₄] have g₅: 1 ≤ p * (p + 1) := by rw [← mul_one (p * (p + 1))] refine Nat.le_mul_of_pos_left ?_ ?_ refine Nat.mul_pos (by linarith) (by linarith) rw [Nat.cast_sub g₅] rw [← sub_eq_add_neg] at h₁ norm_cast norm_cast at h₁ lemma imo_2022_p5_10_12 (p : ℕ) -- (hp : Nat.Prime p) (hp5 : 5 ≤ p) -- (h₀ : (p + 1) ^ 2 ∣ p ^ p - p) -- (gpo : Odd p) -- (gpe : Even (p - 1)) -- (g₁ : (-1) ^ (p - 1) = 1) -- (g₂ : (-1) ^ p = -1) (h₁ : ↑p ^ p - ↑p ≡ ↑p * (↑p + 1) + -1 - ↑p [ZMOD (↑p + 1) ^ 2]) (g₃ : p ≤ p ^ p) : ↑(p ^ p - p) ≡ ↑(p * (p + 1) - 1 - p) [ZMOD ↑(((↑p:ℤ) + 1) ^ 2)] := by have g₄: p ≤ p * (p + 1) - 1 := by rw [mul_add] simp rw [add_comm, Nat.add_sub_assoc] simp rw [← pow_two] refine Nat.one_le_pow 2 p (by linarith) rw [Nat.cast_sub g₄] have g₅: 1 ≤ p * (p + 1) := by rw [← mul_one (p * (p + 1))] refine Nat.le_mul_of_pos_left ?_ ?_ refine Nat.mul_pos (by linarith) (by linarith) rw [Nat.cast_sub g₅] rw [← sub_eq_add_neg] at h₁ norm_cast norm_cast at h₁ lemma imo_2022_p5_10_13 (p : ℕ) -- (hp : Nat.Prime p) (hp5 : 5 ≤ p) -- (h₀ : (p + 1) ^ 2 ∣ p ^ p - p) -- (gpo : Odd p) -- (gpe : Even (p - 1)) -- (g₁ : (-1) ^ (p - 1) = 1) -- (g₂ : (-1) ^ p = -1) (h₁ : ↑p ^ p - ↑p ≡ ↑p * (↑p + 1) + -1 - ↑p [ZMOD (↑p + 1) ^ 2]) (g₃ : p ≤ p ^ p) (g₄ : p ≤ p * (p + 1) - 1) : p ^ p - p ≡ p * (p + 1) - 1 - p [MOD (p + 1) ^ 2] := by refine Int.natCast_modEq_iff.mp ?_ rw [Nat.cast_sub g₃] rw [Nat.cast_sub g₄] have g₅: 1 ≤ p * (p + 1) := by rw [← mul_one (p * (p + 1))] refine Nat.le_mul_of_pos_left ?_ ?_ refine Nat.mul_pos (by linarith) (by linarith) rw [Nat.cast_sub g₅] rw [← sub_eq_add_neg] at h₁ norm_cast norm_cast at h₁ lemma imo_2022_p5_10_14 (p : ℕ) -- (hp : Nat.Prime p) (hp5 : 5 ≤ p) -- (h₀ : (p + 1) ^ 2 ∣ p ^ p - p) -- (gpo : Odd p) -- (gpe : Even (p - 1)) -- (g₁ : (-1) ^ (p - 1) = 1) -- (g₂ : (-1) ^ p = -1) (h₁ : ↑p ^ p - ↑p ≡ ↑p * (↑p + 1) + -1 - ↑p [ZMOD (↑p + 1) ^ 2]) (g₃ : p ≤ p ^ p) : ↑(p ^ p) - ↑p ≡ ↑(p * (p + 1) - 1 - p) [ZMOD ↑(((↑p:ℤ) + 1) ^ 2)] := by have g₄: p ≤ p * (p + 1) - 1 := by rw [mul_add] simp rw [add_comm, Nat.add_sub_assoc] simp rw [← pow_two] refine Nat.one_le_pow 2 p (by linarith) rw [Nat.cast_sub g₄] have g₅: 1 ≤ p * (p + 1) := by rw [← mul_one (p * (p + 1))] refine Nat.le_mul_of_pos_left ?_ ?_ refine Nat.mul_pos (by linarith) (by linarith) rw [Nat.cast_sub g₅] rw [← sub_eq_add_neg] at h₁ norm_cast norm_cast at h₁ lemma imo_2022_p5_10_15 (p : ℕ) -- (hp : Nat.Prime p) (hp5 : 5 ≤ p) : -- (h₀ : (p + 1) ^ 2 ∣ p ^ p - p) -- (gpo : Odd p) -- (gpe : Even (p - 1)) -- (g₁ : (-1) ^ (p - 1) = 1) -- (g₂ : (-1) ^ p = -1) -- (h₁ : ↑p ^ p - ↑p ≡ ↑p * (↑p + 1) + -1 - ↑p [ZMOD (↑p + 1) ^ 2]) -- (g₃ : p ≤ p ^ p) : p ≤ p * (p + 1) - 1 := by rw [mul_add] simp rw [add_comm, Nat.add_sub_assoc] simp rw [← pow_two] refine Nat.one_le_pow 2 p (by linarith) lemma imo_2022_p5_10_16 (p : ℕ) -- (hp : Nat.Prime p) (hp5 : 5 ≤ p) -- (h₀ : (p + 1) ^ 2 ∣ p ^ p - p) -- (gpo : Odd p) -- (gpe : Even (p - 1)) -- (g₁ : (-1) ^ (p - 1) = 1) -- (g₂ : (-1) ^ p = -1) (h₁ : ↑p ^ p - ↑p ≡ ↑p * (↑p + 1) + -1 - ↑p [ZMOD (↑p + 1) ^ 2]) (g₃ : p ≤ p ^ p) (g₄ : p ≤ p * (p + 1) - 1) : ↑(p ^ p) - ↑p ≡ ↑(p * (p + 1) - 1 - p) [ZMOD ↑(((↑p:ℤ) + 1) ^ 2)] := by rw [Nat.cast_sub g₄] have g₅: 1 ≤ p * (p + 1) := by rw [← mul_one (p * (p + 1))] refine Nat.le_mul_of_pos_left ?_ ?_ refine Nat.mul_pos (by linarith) (by linarith) rw [Nat.cast_sub g₅] rw [← sub_eq_add_neg] at h₁ norm_cast norm_cast at h₁ lemma imo_2022_p5_10_17 (p : ℕ) -- (hp : Nat.Prime p) (hp5 : 5 ≤ p) : -- (h₀ : (p + 1) ^ 2 ∣ p ^ p - p) -- (gpo : Odd p) -- (gpe : Even (p - 1)) -- (g₁ : (-1) ^ (p - 1) = 1) -- (g₂ : (-1) ^ p = -1) -- (h₁ : ↑p ^ p - ↑p ≡ ↑p * (↑p + 1) + -1 - ↑p [ZMOD (↑p + 1) ^ 2]) -- (g₃ : p ≤ p ^ p) -- (g₄ : p ≤ p * (p + 1) - 1) : 1 ≤ p * (p + 1) := by rw [← mul_one (p * (p + 1))] refine Nat.le_mul_of_pos_left ?_ ?_ refine Nat.mul_pos (by linarith) (by linarith) lemma imo_2022_p5_10_18 (p : ℕ) -- (hp : Nat.Prime p) (hp5 : 5 ≤ p) : -- (h₀ : (p + 1) ^ 2 ∣ p ^ p - p) -- (gpo : Odd p) -- (gpe : Even (p - 1)) -- (g₁ : (-1) ^ (p - 1) = 1) -- (g₂ : (-1) ^ p = -1) -- (h₁ : ↑p ^ p - ↑p ≡ ↑p * (↑p + 1) + -1 - ↑p [ZMOD (↑p + 1) ^ 2]) -- (g₃ : p ≤ p ^ p) -- (g₄ : p ≤ p * (p + 1) - 1) : 1 ≤ p * (p + 1) - 27 := by have h₂: 6 ≤ (p + 1) := by linarith have h₃: 5 * 6 ≤ p * (p + 1) := by exact Nat.mul_le_mul hp5 h₂ norm_num at h₃ have h₄: 30 - 27 ≤ p * (p + 1) - 27 := by exact Nat.sub_le_sub_right h₃ 27 norm_num at h₄ exact le_trans (by linarith) h₄ lemma imo_2022_p5_10_19 (p : ℕ) -- (hp : Nat.Prime p) -- (hp5 : 5 ≤ p) -- (h₀ : (p + 1) ^ 2 ∣ p ^ p - p) -- (gpo : Odd p) -- (gpe : Even (p - 1)) -- (g₁ : (-1) ^ (p - 1) = 1) -- (g₂ : (-1) ^ p = -1) (h₁ : ↑p ^ p - ↑p ≡ ↑p * (↑p + 1) + -1 - ↑p [ZMOD (↑p + 1) ^ 2]) (g₃ : p ≤ p ^ p) (g₄ : p ≤ p * (p + 1) - 1) (g₅ : 1 ≤ p * (p + 1)) : ↑(p ^ p) - ↑p ≡ ↑(p * (p + 1) - 1) - ↑p [ZMOD ↑(((↑p:ℤ) + 1) ^ 2)] := by rw [Nat.cast_sub g₅] rw [← sub_eq_add_neg] at h₁ norm_cast norm_cast at h₁ lemma imo_2022_p5_10_20 (p : ℕ) -- (hp : Nat.Prime p) (hp5 : 5 ≤ p) (h₀ : (p + 1) ^ 2 ∣ p ^ p - p) (gpo : Odd p) (gpe : Even (p - 1)) (g₁ : (-1) ^ (p - 1) = 1) (g₂ : (-1) ^ p = -1) -- (h₁ : ↑p ^ p - ↑p ≡ ↑p * (↑p + 1) + -1 - ↑p [ZMOD (↑p + 1) ^ 2]) (h₂ : p ^ p - p ≡ p * (p + 1) - 1 - p [MOD (p + 1) ^ 2]) : False := by have h₃: p * (p + 1) - 1 - p = p^2 - 1 := by rw [Nat.sub_sub, mul_add] simp rw [← pow_two] exact Nat.add_sub_add_right (p^2) p 1 rw [h₃] at h₂ clear h₃ gpo gpe g₁ g₂ -- now derive a line of contradictions from h₀ have hc₁: (p ^ p - p) ≡ 0 [MOD (p+1)^2] := by exact modEq_zero_iff_dvd.mpr h₀ -- mix the contradiction with what we had in h₂ have h₄: p ^ 2 - 1 ≡ 0 [MOD (p+1)^2] := by apply Nat.ModEq.symm at h₂ exact Nat.ModEq.trans h₂ hc₁ have h₅: p - 1 ≡ 0 [MOD (p+1)] := by rw [pow_two] at h₄ have g₀: p^2 - 1^2 = (p-1) * (p+1) := by rw [mul_comm] exact Nat.sq_sub_sq p 1 simp at g₀ rw [g₀] at h₄ have g₁: p + 1 ≠ 0 := by linarith refine Nat.ModEq.mul_right_cancel' g₁ ?_ rw [zero_mul] exact h₄ have h₆: p - 1 ≤ 0 := by refine Nat.ModEq.le_of_lt_add h₅ ?_ simp rw [← succ_eq_add_one] refine Nat.sub_lt_succ p 1 have h₇: 0 < p - 1 := by simp linarith linarith [h₆,h₇] lemma imo_2022_p5_10_21 (p : ℕ) : -- (hp : Nat.Prime p) -- (hp5 : 5 ≤ p) -- (h₀ : (p + 1) ^ 2 ∣ p ^ p - p) -- (gpo : Odd p) -- (gpe : Even (p - 1)) -- (g₁ : (-1) ^ (p - 1) = 1) -- (g₂ : (-1) ^ p = -1) -- (h₁ : ↑p ^ p - ↑p ≡ ↑p * (↑p + 1) + -1 - ↑p [ZMOD (↑p + 1) ^ 2]) -- (h₂ : p ^ p - p ≡ p * (p + 1) - 1 - p [MOD (p + 1) ^ 2]) : p * (p + 1) - 1 - p = p ^ 2 - 1 := by rw [Nat.sub_sub, mul_add] simp rw [← pow_two] exact Nat.add_sub_add_right (p^2) p 1 lemma imo_2022_p5_10_22 (p : ℕ) -- (hp : Nat.Prime p) (hp5 : 5 ≤ p) (h₀ : (p + 1) ^ 2 ∣ p ^ p - p) (gpo : Odd p) (gpe : Even (p - 1)) (g₁ : (-1) ^ (p - 1) = 1) (g₂ : (-1) ^ p = -1) -- (h₁ : ↑p ^ p - ↑p ≡ ↑p * (↑p + 1) + -1 - ↑p [ZMOD (↑p + 1) ^ 2]) (h₂ : p ^ p - p ≡ p * (p + 1) - 1 - p [MOD (p + 1) ^ 2]) (h₃ : p * (p + 1) - 1 - p = p ^ 2 - 1) : False := by rw [h₃] at h₂ clear h₃ gpo gpe g₁ g₂ -- now derive a line of contradictions from h₀ have hc₁: (p ^ p - p) ≡ 0 [MOD (p+1)^2] := by exact modEq_zero_iff_dvd.mpr h₀ -- mix the contradiction with what we had in h₂ have h₄: p ^ 2 - 1 ≡ 0 [MOD (p+1)^2] := by apply Nat.ModEq.symm at h₂ exact Nat.ModEq.trans h₂ hc₁ have h₅: p - 1 ≡ 0 [MOD (p+1)] := by rw [pow_two] at h₄ have g₀: p^2 - 1^2 = (p-1) * (p+1) := by rw [mul_comm] exact Nat.sq_sub_sq p 1 simp at g₀ rw [g₀] at h₄ have g₁: p + 1 ≠ 0 := by linarith refine Nat.ModEq.mul_right_cancel' g₁ ?_ rw [zero_mul] exact h₄ have h₆: p - 1 ≤ 0 := by refine Nat.ModEq.le_of_lt_add h₅ ?_ simp rw [← succ_eq_add_one] refine Nat.sub_lt_succ p 1 have h₇: 0 < p - 1 := by simp linarith linarith [h₆,h₇] lemma imo_2022_p5_10_23 (p : ℕ) -- (hp : Nat.Prime p) (hp5 : 5 ≤ p) -- (h₀ : (p + 1) ^ 2 ∣ p ^ p - p) -- (h₁ : ↑p ^ p - ↑p ≡ ↑p * (↑p + 1) + -1 - ↑p [ZMOD (↑p + 1) ^ 2]) (h₂ : p ^ p - p ≡ p ^ 2 - 1 [MOD (p + 1) ^ 2]) (hc₁ : p ^ p - p ≡ 0 [MOD (p + 1) ^ 2]) : False := by -- mix the contradiction with what we had in h₂ have h₄: p ^ 2 - 1 ≡ 0 [MOD (p+1)^2] := by apply Nat.ModEq.symm at h₂ exact Nat.ModEq.trans h₂ hc₁ have h₅: p - 1 ≡ 0 [MOD (p+1)] := by rw [pow_two] at h₄ have g₀: p^2 - 1^2 = (p-1) * (p+1) := by rw [mul_comm] exact Nat.sq_sub_sq p 1 simp at g₀ rw [g₀] at h₄ have g₁: p + 1 ≠ 0 := by linarith refine Nat.ModEq.mul_right_cancel' g₁ ?_ rw [zero_mul] exact h₄ have h₆: p - 1 ≤ 0 := by refine Nat.ModEq.le_of_lt_add h₅ ?_ simp rw [← succ_eq_add_one] refine Nat.sub_lt_succ p 1 have h₇: 0 < p - 1 := by simp linarith linarith [h₆,h₇] lemma imo_2022_p5_10_24 (p : ℕ) -- (hp : Nat.Prime p) -- (hp5 : 5 ≤ p) -- (h₀ : (p + 1) ^ 2 ∣ p ^ p - p) -- (h₁ : ↑p ^ p - ↑p ≡ ↑p * (↑p + 1) + -1 - ↑p [ZMOD (↑p + 1) ^ 2]) (h₂ : p ^ p - p ≡ p ^ 2 - 1 [MOD (p + 1) ^ 2]) (hc₁ : p ^ p - p ≡ 0 [MOD (p + 1) ^ 2]) : p ^ 2 - 1 ≡ 0 [MOD (p + 1) ^ 2] := by apply Nat.ModEq.symm at h₂ exact Nat.ModEq.trans h₂ hc₁ lemma imo_2022_p5_10_25 (p : ℕ) -- (hp : Nat.Prime p) (hp5 : 5 ≤ p) -- (h₀ : (p + 1) ^ 2 ∣ p ^ p - p) -- (h₁ : ↑p ^ p - ↑p ≡ ↑p * (↑p + 1) + -1 - ↑p [ZMOD (↑p + 1) ^ 2]) -- (h₂ : p ^ p - p ≡ p ^ 2 - 1 [MOD (p + 1) ^ 2]) -- (hc₁ : p ^ p - p ≡ 0 [MOD (p + 1) ^ 2]) (h₄ : p ^ 2 - 1 ≡ 0 [MOD (p + 1) ^ 2]) : False := by have h₅: p - 1 ≡ 0 [MOD (p+1)] := by rw [pow_two] at h₄ have g₀: p^2 - 1^2 = (p-1) * (p+1) := by rw [mul_comm] exact Nat.sq_sub_sq p 1 simp at g₀ rw [g₀] at h₄ have g₁: p + 1 ≠ 0 := by linarith refine Nat.ModEq.mul_right_cancel' g₁ ?_ rw [zero_mul] exact h₄ have h₆: p - 1 ≤ 0 := by refine Nat.ModEq.le_of_lt_add h₅ ?_ simp rw [← succ_eq_add_one] refine Nat.sub_lt_succ p 1 have h₇: 0 < p - 1 := by simp linarith linarith [h₆,h₇] lemma imo_2022_p5_10_26 (p : ℕ) -- (hp : Nat.Prime p) (hp5 : 5 ≤ p) -- (h₀ : (p + 1) ^ 2 ∣ p ^ p - p) -- (h₁ : ↑p ^ p - ↑p ≡ ↑p * (↑p + 1) + -1 - ↑p [ZMOD (↑p + 1) ^ 2]) -- (h₂ : p ^ p - p ≡ p ^ 2 - 1 [MOD (p + 1) ^ 2]) -- (hc₁ : p ^ p - p ≡ 0 [MOD (p + 1) ^ 2]) (h₄ : p ^ 2 - 1 ≡ 0 [MOD (p + 1) ^ 2]) : p - 1 ≡ 0 [MOD p + 1] := by rw [pow_two] at h₄ have g₀: p^2 - 1^2 = (p-1) * (p+1) := by rw [mul_comm] exact Nat.sq_sub_sq p 1 simp at g₀ rw [g₀] at h₄ have g₁: p + 1 ≠ 0 := by linarith refine Nat.ModEq.mul_right_cancel' g₁ ?_ rw [zero_mul] exact h₄ lemma imo_2022_p5_10_27 (p : ℕ) : -- (hp : Nat.Prime p) -- (hp5 : 5 ≤ p) -- (h₀ : (p + 1) ^ 2 ∣ p ^ p - p) -- (h₁ : ↑p ^ p - ↑p ≡ ↑p * (↑p + 1) + -1 - ↑p [ZMOD (↑p + 1) ^ 2]) -- (h₂ : p ^ p - p ≡ p ^ 2 - 1 [MOD (p + 1) ^ 2]) -- (hc₁ : p ^ p - p ≡ 0 [MOD (p + 1) ^ 2]) -- (h₄ : p ^ 2 - 1 ≡ 0 [MOD (p + 1) * (p + 1)]) : p ^ 2 - 1 ^ 2 = (p - 1) * (p + 1) := by rw [mul_comm] exact Nat.sq_sub_sq p 1 lemma imo_2022_p5_10_28 (p : ℕ) -- (hp : Nat.Prime p) (hp5 : 5 ≤ p) -- (h₀ : (p + 1) ^ 2 ∣ p ^ p - p) -- (h₁ : ↑p ^ p - ↑p ≡ ↑p * (↑p + 1) + -1 - ↑p [ZMOD (↑p + 1) ^ 2]) -- (h₂ : p ^ p - p ≡ p ^ 2 - 1 [MOD (p + 1) ^ 2]) -- (hc₁ : p ^ p - p ≡ 0 [MOD (p + 1) ^ 2]) (h₄ : p ^ 2 - 1 ≡ 0 [MOD (p + 1) * (p + 1)]) (g₀ : p ^ 2 - 1 ^ 2 = (p - 1) * (p + 1)) : p - 1 ≡ 0 [MOD p + 1] := by simp at g₀ rw [g₀] at h₄ have g₁: p + 1 ≠ 0 := by linarith refine Nat.ModEq.mul_right_cancel' g₁ ?_ rw [zero_mul] exact h₄ lemma imo_2022_p5_10_29 (p : ℕ) -- (hp : Nat.Prime p) -- (hp5 : 5 ≤ p) -- (h₀ : (p + 1) ^ 2 ∣ p ^ p - p) -- (h₁ : ↑p ^ p - ↑p ≡ ↑p * (↑p + 1) + -1 - ↑p [ZMOD (↑p + 1) ^ 2]) -- (h₂ : p ^ p - p ≡ p ^ 2 - 1 [MOD (p + 1) ^ 2]) -- (hc₁ : p ^ p - p ≡ 0 [MOD (p + 1) ^ 2]) (h₄ : p ^ 2 - 1 ≡ 0 [MOD (p + 1) ^ 2]) : (p - 1) * (p + 1) ≡ 0 [MOD (p + 1) * (p + 1)] := by rw [pow_two] at h₄ have g₀: p^2 - 1^2 = (p-1) * (p+1) := by rw [mul_comm] exact Nat.sq_sub_sq p 1 simp at g₀ rw [g₀] at h₄ exact h₄ lemma imo_2022_p5_10_30 (p : ℕ) -- (hp : Nat.Prime p) -- (hp5 : 5 ≤ p) -- (h₀ : (p + 1) ^ 2 ∣ p ^ p - p) -- (h₁ : ↑p ^ p - ↑p ≡ ↑p * (↑p + 1) + -1 - ↑p [ZMOD (↑p + 1) ^ 2]) -- (h₂ : p ^ p - p ≡ p ^ 2 - 1 [MOD (p + 1) ^ 2]) -- (hc₁ : p ^ p - p ≡ 0 [MOD (p + 1) ^ 2]) (h₄ : (p - 1) * (p + 1) ≡ 0 [MOD (p + 1) * (p + 1)]) -- (g₀ : p ^ 2 - 1 = (p - 1) * (p + 1)) (g₁ : p + 1 ≠ 0) : p - 1 ≡ 0 [MOD p + 1] := by refine Nat.ModEq.mul_right_cancel' g₁ ?_ rw [zero_mul] exact h₄ lemma imo_2022_p5_10_31 (p : ℕ) -- (hp : Nat.Prime p) (hp5 : 5 ≤ p) -- (h₀ : (p + 1) ^ 2 ∣ p ^ p - p) -- (h₁ : ↑p ^ p - ↑p ≡ ↑p * (↑p + 1) + -1 - ↑p [ZMOD (↑p + 1) ^ 2]) -- (h₂ : p ^ p - p ≡ p ^ 2 - 1 [MOD (p + 1) ^ 2]) -- (hc₁ : p ^ p - p ≡ 0 [MOD (p + 1) ^ 2]) -- (h₄ : p ^ 2 - 1 ≡ 0 [MOD (p + 1) ^ 2]) (h₅ : p - 1 ≡ 0 [MOD p + 1]) : False := by have h₆: p - 1 ≤ 0 := by refine Nat.ModEq.le_of_lt_add h₅ ?_ simp rw [← succ_eq_add_one] refine Nat.sub_lt_succ p 1 have h₇: 0 < p - 1 := by simp linarith linarith [h₆,h₇] lemma imo_2022_p5_10_32 (p : ℕ) -- (hp : Nat.Prime p) -- (hp5 : 5 ≤ p) -- (h₀ : (p + 1) ^ 2 ∣ p ^ p - p) -- (h₁ : ↑p ^ p - ↑p ≡ ↑p * (↑p + 1) + -1 - ↑p [ZMOD (↑p + 1) ^ 2]) -- (h₂ : p ^ p - p ≡ p ^ 2 - 1 [MOD (p + 1) ^ 2]) -- (hc₁ : p ^ p - p ≡ 0 [MOD (p + 1) ^ 2]) -- (h₄ : p ^ 2 - 1 ≡ 0 [MOD (p + 1) ^ 2]) (h₅ : p - 1 ≡ 0 [MOD p + 1]) : p - 1 ≤ 0 := by refine Nat.ModEq.le_of_lt_add h₅ ?_ simp rw [← succ_eq_add_one] refine Nat.sub_lt_succ p 1 lemma imo_2022_p5_10_33 (p : ℕ) : -- (hp : Nat.Prime p) -- (hp5 : 5 ≤ p) -- (h₀ : (p + 1) ^ 2 ∣ p ^ p - p) -- (h₁ : ↑p ^ p - ↑p ≡ ↑p * (↑p + 1) + -1 - ↑p [ZMOD (↑p + 1) ^ 2]) -- (h₂ : p ^ p - p ≡ p ^ 2 - 1 [MOD (p + 1) ^ 2]) -- (hc₁ : p ^ p - p ≡ 0 [MOD (p + 1) ^ 2]) -- (h₄ : p ^ 2 - 1 ≡ 0 [MOD (p + 1) ^ 2]) -- (h₅ : p - 1 ≡ 0 [MOD p + 1]) : p - 1 < 0 + (p + 1) := by simp rw [← succ_eq_add_one] refine Nat.sub_lt_succ p 1 lemma imo_2022_p5_10_34 (p : ℕ) -- (hp : Nat.Prime p) (hp5 : 5 ≤ p) -- (h₀ : (p + 1) ^ 2 ∣ p ^ p - p) -- (h₁ : ↑p ^ p - ↑p ≡ ↑p * (↑p + 1) + -1 - ↑p [ZMOD (↑p + 1) ^ 2]) -- (h₂ : p ^ p - p ≡ p ^ 2 - 1 [MOD (p + 1) ^ 2]) -- (hc₁ : p ^ p - p ≡ 0 [MOD (p + 1) ^ 2]) -- (h₄ : p ^ 2 - 1 ≡ 0 [MOD (p + 1) ^ 2]) -- (h₅ : p - 1 ≡ 0 [MOD p + 1]) (h₆ : p - 1 ≤ 0) : False := by have h₇: 0 < p - 1 := by simp linarith linarith [h₆,h₇] lemma imo_2022_p5_11 (p: ℕ) -- (hp: Nat.Prime p) (hpl: 5 ≤ p) : (p + p.factorial < p ^ p) := by -- induction p using Nat.case_strong_induction_on with n ih, refine Nat.le_induction ?_ ?_ p (hpl) . exact Nat.lt_of_sub_eq_succ rfl . intros n hn h₁ have h₂: n + 1 + (n + 1).factorial = (n.factorial + 1) * (n + 1) := by rw[add_mul, one_mul, Nat.factorial_succ] rw [add_comm (n + 1)] rw [mul_comm (n + 1)] rw [h₂, pow_add, pow_one ] refine Nat.mul_lt_mul_of_pos_right ?_ (by linarith) have h₅: n ^ n < (n + 1) ^ n := by refine Nat.pow_lt_pow_left ?_ ?_ . exact lt_add_one n . refine Nat.ne_of_gt ?_ linarith linarith lemma imo_2022_p5_11_1 : -- (p : ℕ) -- (hpl : 5 ≤ p) : ∀ (n : ℕ), 5 ≤ n → n + n ! < n ^ n → n + 1 + (n + 1)! < (n + 1) ^ (n + 1) := by intros n hn h₁ have h₂: n + 1 + (n + 1).factorial = (n.factorial + 1) * (n + 1) := by rw[add_mul, one_mul, Nat.factorial_succ] rw [add_comm (n + 1)] rw [mul_comm (n + 1)] rw [h₂, pow_add, pow_one ] refine Nat.mul_lt_mul_of_pos_right ?_ (by linarith) have h₅: n ^ n < (n + 1) ^ n := by refine Nat.pow_lt_pow_left ?_ ?_ . exact lt_add_one n . refine Nat.ne_of_gt ?_ linarith linarith lemma imo_2022_p5_11_2 -- (p : ℕ) -- (hpl : 5 ≤ p) (n : ℕ) (hn : 5 ≤ n) (h₁ : n + n ! < n ^ n) : n + 1 + (n + 1)! < (n + 1) ^ (n + 1) := by have h₂: n + 1 + (n + 1).factorial = (n.factorial + 1) * (n + 1) := by rw[add_mul, one_mul, Nat.factorial_succ] rw [add_comm (n + 1)] rw [mul_comm (n + 1)] rw [h₂, pow_add, pow_one ] refine Nat.mul_lt_mul_of_pos_right ?_ (by linarith) have h₅: n ^ n < (n + 1) ^ n := by refine Nat.pow_lt_pow_left ?_ ?_ . exact lt_add_one n . refine Nat.ne_of_gt ?_ linarith linarith lemma imo_2022_p5_11_3 -- (p : ℕ) -- (hpl : 5 ≤ p) (n : ℕ) : -- (hn : 5 ≤ n) -- (h₁ : n + n ! < n ^ n) : n + 1 + (n + 1)! = (n ! + 1) * (n + 1) := by rw[add_mul, one_mul, Nat.factorial_succ] rw [add_comm (n + 1)] rw [mul_comm (n + 1)] lemma imo_2022_p5_11_4 -- (p : ℕ) -- (hpl : 5 ≤ p) (n : ℕ) (hn : 5 ≤ n) (h₁ : n + n ! < n ^ n) (h₂ : n + 1 + (n + 1)! = (n ! + 1) * (n + 1)) : n + 1 + (n + 1)! < (n + 1) ^ (n + 1) := by rw [h₂, pow_add, pow_one ] refine Nat.mul_lt_mul_of_pos_right ?_ (by linarith) have h₅: n ^ n < (n + 1) ^ n := by refine Nat.pow_lt_pow_left ?_ ?_ . exact lt_add_one n . refine Nat.ne_of_gt ?_ linarith linarith lemma imo_2022_p5_11_5 -- (p : ℕ) -- (hpl : 5 ≤ p) (n : ℕ) (hn : 5 ≤ n) (h₁ : n + n ! < n ^ n) : -- (h₂ : n + 1 + (n + 1)! = (n ! + 1) * (n + 1)) : n ! + 1 < (n + 1) ^ n := by have h₅: n ^ n < (n + 1) ^ n := by refine Nat.pow_lt_pow_left ?_ ?_ . exact lt_add_one n . refine Nat.ne_of_gt ?_ linarith linarith lemma imo_2022_p5_11_6 -- (p : ℕ) -- (hpl : 5 ≤ p) (n : ℕ) (hn : 5 ≤ n) -- (h₁ : n + n ! < n ^ n) -- (h₂ : n + 1 + (n + 1)! = (n ! + 1) * (n + 1)) (h₄ : n + n ! < n ^ n) : n ! + 1 < (n + 1) ^ n := by have h₅: n ^ n < (n + 1) ^ n := by refine Nat.pow_lt_pow_left ?_ ?_ . exact lt_add_one n . refine Nat.ne_of_gt ?_ linarith linarith lemma imo_2022_p5_11_7 -- (p : ℕ) -- (hpl : 5 ≤ p) (n : ℕ) (hn : 5 ≤ n) : -- (h₁ : n + n ! < n ^ n) -- (h₂ : n + 1 + (n + 1)! = (n ! + 1) * (n + 1)) -- (h₄ : n + n ! < n ^ n) : n ^ n < (n + 1) ^ n := by refine Nat.pow_lt_pow_left ?_ ?_ . exact lt_add_one n . refine Nat.ne_of_gt ?_ linarith lemma imo_2022_p5_12 (b p: ℕ) (hp: Nat.Prime p) (hbp: p ≤ b) (h₁: p ^ p = b.factorial + p) (hp5: 5 ≤ p) : (False) := by -- first prove that b = p cannot be by_cases h₄: b = p . exfalso rw [h₄] at h₁ have h₅: p + p.factorial < p^p := by exact imo_2022_p5_11 p hp5 linarith . have hpb: p < b := by exact lt_of_le_of_ne' hbp h₄ clear hbp h₄ have h₂: (p + 1) ^ 2 ∣ b.factorial := by have g₁: p + 1 ≤ b := by exact succ_le_iff.mpr hpb have g₂: 2 ∣ (p + 1) := by have gg₁: Odd p := by refine hp.odd_of_ne_two ?_ linarith have gg₂: Even (p + 1) := by refine gg₁.add_odd ?_ norm_num exact even_iff_two_dvd.mp gg₂ have g₃: 2 * ((p+1)/2) * (p + 1) ∣ b.factorial := by have gg₁: (p + 1).factorial ∣ b.factorial := by exact Nat.factorial_dvd_factorial g₁ have gg₂: (p + 1).factorial = (p + 1) * p.factorial := by exact factorial_succ p rw [mul_comm] at gg₂ have gg₃: 6/2 ≤ (p + 1)/2 := by refine Nat.div_le_div_right ?_ linarith norm_num at gg₃ have gg₄: 2 + (p+1)/2 ≤ p := by -- strong induction refine Nat.le_induction ?_ ?_ p (hp5) . norm_num . intros n _ h₂ ring_nf have ggg₁: (n / 2).succ ≤ (n + 1) / 2 + 1 := by rw [← succ_eq_add_one] refine Nat.succ_le_succ ?_ refine Nat.div_le_div_right ?_ linarith simp nth_rewrite 1 [← mul_one 2] rw [Nat.two_mul 1, add_assoc] refine Nat.add_le_add_left ?_ 1 refine le_trans ?_ h₂ rw [add_comm 2 _] nth_rewrite 3 [← mul_one 2] rw [Nat.two_mul 1, ← add_assoc, add_comm 1] exact Nat.add_le_add_right ggg₁ 1 have gg₅: (2+(p+1)/2).factorial ∣ p.factorial := by exact factorial_dvd_factorial gg₄ have gg₆: (2:ℕ).factorial * ((p+1)/2).factorial ∣ p.factorial := by refine dvd_trans ?_ gg₅ exact (2:ℕ).factorial_mul_factorial_dvd_factorial_add ((p + 1) / 2) have gg₇: 2 * ((p+1)/2) ∣ p.factorial := by refine dvd_trans ?_ gg₆ simp refine mul_dvd_mul_left 2 ?_ refine Nat.dvd_factorial (by linarith[gg₃]) (by linarith) have gg₈: 2 * ((p+1)/2) * (p + 1) ∣ p.factorial * (p + 1) := by refine mul_dvd_mul_right ?_ (p + 1) exact gg₇ rw [gg₂] at gg₁ exact dvd_trans gg₈ gg₁ have g₄: 2 * ((p+1)/2) = (p + 1) := by exact Nat.mul_div_cancel' g₂ rw [g₄] at g₃ ring_nf at * exact g₃ have h₃: b.factorial = p ^ p - p := by exact eq_tsub_of_add_eq (h₁.symm) rw [h₃] at h₂ exact imo_2022_p5_10 p hp hp5 h₂ lemma imo_2022_p5_12_1 (b p : ℕ) -- (hp : Nat.Prime p) -- (hbp : p ≤ b) (h₁ : p ^ p = b ! + p) (hp5 : 5 ≤ p) (h₄ : b = p) : False := by rw [h₄] at h₁ have h₅: p + p.factorial < p ^ p := by exact imo_2022_p5_11 p hp5 linarith lemma imo_2022_p5_12_2 (b p : ℕ) -- (hp : Nat.Prime p) -- (hbp : p ≤ b) (h₁ : p ^ p = b ! + p) (hp5 : 5 ≤ p) (h₄ : b = p) (h₅ : p + p ! < p ^ p) : False := by rw [h₄] at h₁ linarith lemma imo_2022_p5_12_3 (b p : ℕ) -- (hp : Nat.Prime p) -- (hbp : p ≤ b) (h₁ : p ^ p = b ! + p) -- (hp5 : 5 ≤ p) (h₄ : b = p) (h₅ : p + p ! < p ^ p) : False := by rw [h₁, add_comm, h₄] at h₅ apply Nat.add_lt_add_iff_right.mp at h₅ linarith lemma imo_2022_p5_12_4 (b p : ℕ) (hp : Nat.Prime p) (h₁ : p ^ p = b ! + p) (hp5 : 5 ≤ p) (hpb : p < b) : False := by have h₂: (p + 1) ^ 2 ∣ b.factorial := by have g₁: p + 1 ≤ b := by exact succ_le_iff.mpr hpb have g₂: 2 ∣ (p + 1) := by have gg₁: Odd p := by refine hp.odd_of_ne_two ?_ linarith have gg₂: Even (p + 1) := by refine gg₁.add_odd ?_ norm_num exact even_iff_two_dvd.mp gg₂ have g₃: 2 * ((p+1)/2) * (p + 1) ∣ b.factorial := by have gg₁: (p + 1).factorial ∣ b.factorial := by exact Nat.factorial_dvd_factorial g₁ have gg₂: (p + 1).factorial = (p + 1) * p.factorial := by exact factorial_succ p rw [mul_comm] at gg₂ have gg₃: 6/2 ≤ (p + 1)/2 := by refine Nat.div_le_div_right ?_ linarith norm_num at gg₃ have gg₄: 2 + (p+1)/2 ≤ p := by -- strong induction refine Nat.le_induction ?_ ?_ p (hp5) . norm_num . intros n _ h₂ ring_nf have ggg₁: (n / 2).succ ≤ (n + 1) / 2 + 1 := by rw [← succ_eq_add_one] refine Nat.succ_le_succ ?_ refine Nat.div_le_div_right ?_ linarith simp nth_rewrite 1 [← mul_one 2] rw [Nat.two_mul 1, add_assoc] refine Nat.add_le_add_left ?_ 1 refine le_trans ?_ h₂ rw [add_comm 2 _] nth_rewrite 3 [← mul_one 2] rw [Nat.two_mul 1, ← add_assoc, add_comm 1] exact Nat.add_le_add_right ggg₁ 1 have gg₅: (2+(p+1)/2).factorial ∣ p.factorial := by exact factorial_dvd_factorial gg₄ have gg₆: (2:ℕ).factorial * ((p+1)/2).factorial ∣ p.factorial := by refine dvd_trans ?_ gg₅ exact (2:ℕ).factorial_mul_factorial_dvd_factorial_add ((p + 1) / 2) have gg₇: 2 * ((p+1)/2) ∣ p.factorial := by refine dvd_trans ?_ gg₆ simp refine mul_dvd_mul_left 2 ?_ refine Nat.dvd_factorial (by linarith[gg₃]) (by linarith) have gg₈: 2 * ((p+1)/2) * (p + 1) ∣ p.factorial * (p + 1) := by refine mul_dvd_mul_right ?_ (p + 1) exact gg₇ rw [gg₂] at gg₁ exact dvd_trans gg₈ gg₁ have g₄: 2 * ((p+1)/2) = (p + 1) := by exact Nat.mul_div_cancel' g₂ rw [g₄] at g₃ ring_nf at * exact g₃ have h₃: b.factorial = p ^ p - p := by exact eq_tsub_of_add_eq (h₁.symm) rw [h₃] at h₂ exact imo_2022_p5_10 p hp hp5 h₂ lemma imo_2022_p5_12_5 (b p : ℕ) (hp : Nat.Prime p) (h₁ : p ^ p = b ! + p) (hp5 : 5 ≤ p) (hpb : p < b) : (p + 1) ^ 2 ∣ b ! := by have g₁: p + 1 ≤ b := by exact succ_le_iff.mpr hpb have g₂: 2 ∣ (p + 1) := by have gg₁: Odd p := by refine hp.odd_of_ne_two ?_ linarith have gg₂: Even (p + 1) := by refine gg₁.add_odd ?_ norm_num exact even_iff_two_dvd.mp gg₂ have g₃: 2 * ((p+1)/2) * (p + 1) ∣ b.factorial := by have gg₁: (p + 1).factorial ∣ b.factorial := by exact Nat.factorial_dvd_factorial g₁ have gg₂: (p + 1).factorial = (p + 1) * p.factorial := by exact factorial_succ p rw [mul_comm] at gg₂ have gg₃: 6/2 ≤ (p + 1)/2 := by refine Nat.div_le_div_right ?_ linarith norm_num at gg₃ have gg₄: 2 + (p+1)/2 ≤ p := by -- strong induction refine Nat.le_induction ?_ ?_ p (hp5) . norm_num . intros n _ h₂ ring_nf have ggg₁: (n / 2).succ ≤ (n + 1) / 2 + 1 := by rw [← succ_eq_add_one] refine Nat.succ_le_succ ?_ refine Nat.div_le_div_right ?_ linarith simp nth_rewrite 1 [← mul_one 2] rw [Nat.two_mul 1, add_assoc] refine Nat.add_le_add_left ?_ 1 refine le_trans ?_ h₂ rw [add_comm 2 _] nth_rewrite 3 [← mul_one 2] rw [Nat.two_mul 1, ← add_assoc, add_comm 1] exact Nat.add_le_add_right ggg₁ 1 have gg₅: (2+(p+1)/2).factorial ∣ p.factorial := by exact factorial_dvd_factorial gg₄ have gg₆: (2:ℕ).factorial * ((p+1)/2).factorial ∣ p.factorial := by refine dvd_trans ?_ gg₅ exact (2:ℕ).factorial_mul_factorial_dvd_factorial_add ((p + 1) / 2) have gg₇: 2 * ((p+1)/2) ∣ p.factorial := by refine dvd_trans ?_ gg₆ simp refine mul_dvd_mul_left 2 ?_ refine Nat.dvd_factorial (by linarith[gg₃]) (by linarith) have gg₈: 2 * ((p+1)/2) * (p + 1) ∣ p.factorial * (p + 1) := by refine mul_dvd_mul_right ?_ (p + 1) exact gg₇ rw [gg₂] at gg₁ exact dvd_trans gg₈ gg₁ have g₄: 2 * ((p+1)/2) = (p + 1) := by exact Nat.mul_div_cancel' g₂ rw [g₄] at g₃ ring_nf at * exact g₃ lemma imo_2022_p5_12_6 (b p : ℕ) (hp : Nat.Prime p) (h₁ : p ^ p = b ! + p) (hp5 : 5 ≤ p) -- (hpb : p < b) (g₁ : p + 1 ≤ b) : (p + 1) ^ 2 ∣ b ! := by have g₂: 2 ∣ (p + 1) := by have gg₁: Odd p := by refine hp.odd_of_ne_two ?_ linarith have gg₂: Even (p + 1) := by refine gg₁.add_odd ?_ norm_num exact even_iff_two_dvd.mp gg₂ have g₃: 2 * ((p+1)/2) * (p + 1) ∣ b.factorial := by have gg₁: (p + 1).factorial ∣ b.factorial := by exact Nat.factorial_dvd_factorial g₁ have gg₂: (p + 1).factorial = (p + 1) * p.factorial := by exact factorial_succ p rw [mul_comm] at gg₂ have gg₃: 6/2 ≤ (p + 1)/2 := by refine Nat.div_le_div_right ?_ linarith norm_num at gg₃ have gg₄: 2 + (p+1)/2 ≤ p := by -- strong induction refine Nat.le_induction ?_ ?_ p (hp5) . norm_num . intros n _ h₂ ring_nf have ggg₁: (n / 2).succ ≤ (n + 1) / 2 + 1 := by rw [← succ_eq_add_one] refine Nat.succ_le_succ ?_ refine Nat.div_le_div_right ?_ linarith simp nth_rewrite 1 [← mul_one 2] rw [Nat.two_mul 1, add_assoc] refine Nat.add_le_add_left ?_ 1 refine le_trans ?_ h₂ rw [add_comm 2 _] nth_rewrite 3 [← mul_one 2] rw [Nat.two_mul 1, ← add_assoc, add_comm 1] exact Nat.add_le_add_right ggg₁ 1 have gg₅: (2+(p+1)/2).factorial ∣ p.factorial := by exact factorial_dvd_factorial gg₄ have gg₆: (2:ℕ).factorial * ((p+1)/2).factorial ∣ p.factorial := by refine dvd_trans ?_ gg₅ exact (2:ℕ).factorial_mul_factorial_dvd_factorial_add ((p + 1) / 2) have gg₇: 2 * ((p+1)/2) ∣ p.factorial := by refine dvd_trans ?_ gg₆ simp refine mul_dvd_mul_left 2 ?_ refine Nat.dvd_factorial (by linarith[gg₃]) (by linarith) have gg₈: 2 * ((p+1)/2) * (p + 1) ∣ p.factorial * (p + 1) := by refine mul_dvd_mul_right ?_ (p + 1) exact gg₇ rw [gg₂] at gg₁ exact dvd_trans gg₈ gg₁ have g₄: 2 * ((p+1)/2) = (p + 1) := by exact Nat.mul_div_cancel' g₂ rw [g₄] at g₃ ring_nf at * exact g₃ lemma imo_2022_p5_12_7 -- (b : ℕ) (p : ℕ) (hp : Nat.Prime p) -- (h₁ : p ^ p = b ! + p) (hp5 : 5 ≤ p) : -- (hpb : p < b) -- (g₁ : p + 1 ≤ b) : 2 ∣ p + 1 := by have gg₁: Odd p := by refine hp.odd_of_ne_two ?_ linarith have gg₂: Even (p + 1) := by refine gg₁.add_odd ?_ norm_num exact even_iff_two_dvd.mp gg₂ lemma imo_2022_p5_12_8 (b p : ℕ) -- (hp : Nat.Prime p) (h₁ : p ^ p = b ! + p) (hp5 : 5 ≤ p) -- (hpb : p < b) (g₁ : p + 1 ≤ b) (g₂ : 2 ∣ p + 1) : (p + 1) ^ 2 ∣ b ! := by have g₃: 2 * ((p+1)/2) * (p + 1) ∣ b.factorial := by have gg₁: (p + 1).factorial ∣ b.factorial := by exact Nat.factorial_dvd_factorial g₁ have gg₂: (p + 1).factorial = (p + 1) * p.factorial := by exact factorial_succ p rw [mul_comm] at gg₂ have gg₃: 6/2 ≤ (p + 1)/2 := by refine Nat.div_le_div_right ?_ linarith norm_num at gg₃ have gg₄: 2 + (p+1)/2 ≤ p := by -- strong induction refine Nat.le_induction ?_ ?_ p (hp5) . norm_num . intros n _ h₂ ring_nf have ggg₁: (n / 2).succ ≤ (n + 1) / 2 + 1 := by rw [← succ_eq_add_one] refine Nat.succ_le_succ ?_ refine Nat.div_le_div_right ?_ linarith simp nth_rewrite 1 [← mul_one 2] rw [Nat.two_mul 1, add_assoc] refine Nat.add_le_add_left ?_ 1 refine le_trans ?_ h₂ rw [add_comm 2 _] nth_rewrite 3 [← mul_one 2] rw [Nat.two_mul 1, ← add_assoc, add_comm 1] exact Nat.add_le_add_right ggg₁ 1 have gg₅: (2+(p+1)/2).factorial ∣ p.factorial := by exact factorial_dvd_factorial gg₄ have gg₆: (2:ℕ).factorial * ((p+1)/2).factorial ∣ p.factorial := by refine dvd_trans ?_ gg₅ exact (2:ℕ).factorial_mul_factorial_dvd_factorial_add ((p + 1) / 2) have gg₇: 2 * ((p+1)/2) ∣ p.factorial := by refine dvd_trans ?_ gg₆ simp refine mul_dvd_mul_left 2 ?_ refine Nat.dvd_factorial (by linarith[gg₃]) (by linarith) have gg₈: 2 * ((p+1)/2) * (p + 1) ∣ p.factorial * (p + 1) := by refine mul_dvd_mul_right ?_ (p + 1) exact gg₇ rw [gg₂] at gg₁ exact dvd_trans gg₈ gg₁ have g₄: 2 * ((p+1)/2) = (p + 1) := by exact Nat.mul_div_cancel' g₂ rw [g₄] at g₃ ring_nf at * exact g₃ lemma imo_2022_p5_12_9 (b p : ℕ) -- (hp : Nat.Prime p) -- (h₁ : p ^ p = b ! + p) (hp5 : 5 ≤ p) -- (hpb : p < b) (g₁ : p + 1 ≤ b) : -- (g₂ : 2 ∣ p + 1) : 2 * ((p + 1) / 2) * (p + 1) ∣ b ! := by have gg₁: (p + 1).factorial ∣ b.factorial := by exact Nat.factorial_dvd_factorial g₁ have gg₂: (p + 1).factorial = (p + 1) * p.factorial := by exact factorial_succ p rw [mul_comm] at gg₂ have gg₃: 6/2 ≤ (p + 1)/2 := by refine Nat.div_le_div_right ?_ linarith norm_num at gg₃ have gg₄: 2 + (p+1)/2 ≤ p := by -- strong induction refine Nat.le_induction ?_ ?_ p (hp5) . norm_num . intros n _ h₂ ring_nf have ggg₁: (n / 2).succ ≤ (n + 1) / 2 + 1 := by rw [← succ_eq_add_one] refine Nat.succ_le_succ ?_ refine Nat.div_le_div_right ?_ linarith simp nth_rewrite 1 [← mul_one 2] rw [Nat.two_mul 1, add_assoc] refine Nat.add_le_add_left ?_ 1 refine le_trans ?_ h₂ rw [add_comm 2 _] nth_rewrite 3 [← mul_one 2] rw [Nat.two_mul 1, ← add_assoc, add_comm 1] exact Nat.add_le_add_right ggg₁ 1 have gg₅: (2+(p+1)/2).factorial ∣ p.factorial := by exact factorial_dvd_factorial gg₄ have gg₆: (2:ℕ).factorial * ((p+1)/2).factorial ∣ p.factorial := by refine dvd_trans ?_ gg₅ exact (2:ℕ).factorial_mul_factorial_dvd_factorial_add ((p + 1) / 2) have gg₇: 2 * ((p+1)/2) ∣ p.factorial := by refine dvd_trans ?_ gg₆ simp refine mul_dvd_mul_left 2 ?_ refine Nat.dvd_factorial (by linarith[gg₃]) (by linarith) have gg₈: 2 * ((p+1)/2) * (p + 1) ∣ p.factorial * (p + 1) := by refine mul_dvd_mul_right ?_ (p + 1) exact gg₇ rw [gg₂] at gg₁ exact dvd_trans gg₈ gg₁ lemma imo_2022_p5_12_10 (b p : ℕ) -- (hp : Nat.Prime p) -- (h₁ : p ^ p = b ! + p) (hp5 : 5 ≤ p) -- (hpb : p < b) -- (g₁ : p + 1 ≤ b) -- (g₂ : 2 ∣ p + 1) (gg₁ : (p + 1)! ∣ b !) : 2 * ((p + 1) / 2) * (p + 1) ∣ b ! := by have gg₂: (p + 1).factorial = (p + 1) * p.factorial := by exact factorial_succ p rw [mul_comm] at gg₂ have gg₃: 6/2 ≤ (p + 1)/2 := by refine Nat.div_le_div_right ?_ linarith norm_num at gg₃ have gg₄: 2 + (p+1)/2 ≤ p := by -- strong induction refine Nat.le_induction ?_ ?_ p (hp5) . norm_num . intros n _ h₂ ring_nf have ggg₁: (n / 2).succ ≤ (n + 1) / 2 + 1 := by rw [← succ_eq_add_one] refine Nat.succ_le_succ ?_ refine Nat.div_le_div_right ?_ linarith simp nth_rewrite 1 [← mul_one 2] rw [Nat.two_mul 1, add_assoc] refine Nat.add_le_add_left ?_ 1 refine le_trans ?_ h₂ rw [add_comm 2 _] nth_rewrite 3 [← mul_one 2] rw [Nat.two_mul 1, ← add_assoc, add_comm 1] exact Nat.add_le_add_right ggg₁ 1 have gg₅: (2+(p+1)/2).factorial ∣ p.factorial := by exact factorial_dvd_factorial gg₄ have gg₆: (2:ℕ).factorial * ((p+1)/2).factorial ∣ p.factorial := by refine dvd_trans ?_ gg₅ exact (2:ℕ).factorial_mul_factorial_dvd_factorial_add ((p + 1) / 2) have gg₇: 2 * ((p+1)/2) ∣ p.factorial := by refine dvd_trans ?_ gg₆ simp refine mul_dvd_mul_left 2 ?_ refine Nat.dvd_factorial (by linarith[gg₃]) (by linarith) have gg₈: 2 * ((p+1)/2) * (p + 1) ∣ p.factorial * (p + 1) := by refine mul_dvd_mul_right ?_ (p + 1) exact gg₇ rw [gg₂] at gg₁ exact dvd_trans gg₈ gg₁ lemma imo_2022_p5_12_11 (b p : ℕ) -- (hp : Nat.Prime p) -- (h₁ : p ^ p = b ! + p) (hp5 : 5 ≤ p) -- (hpb : p < b) -- (g₁ : p + 1 ≤ b) -- (g₂ : 2 ∣ p + 1) (gg₁ : (p + 1)! ∣ b !) (gg₂ : (p + 1)! = (p + 1) * p !) : 2 * ((p + 1) / 2) * (p + 1) ∣ b ! := by rw [mul_comm] at gg₂ have gg₃: 6/2 ≤ (p + 1)/2 := by refine Nat.div_le_div_right ?_ linarith norm_num at gg₃ have gg₄: 2 + (p+1)/2 ≤ p := by -- strong induction refine Nat.le_induction ?_ ?_ p (hp5) . norm_num . intros n _ h₂ ring_nf have ggg₁: (n / 2).succ ≤ (n + 1) / 2 + 1 := by rw [← succ_eq_add_one] refine Nat.succ_le_succ ?_ refine Nat.div_le_div_right ?_ linarith simp nth_rewrite 1 [← mul_one 2] rw [Nat.two_mul 1, add_assoc] refine Nat.add_le_add_left ?_ 1 refine le_trans ?_ h₂ rw [add_comm 2 _] nth_rewrite 3 [← mul_one 2] rw [Nat.two_mul 1, ← add_assoc, add_comm 1] exact Nat.add_le_add_right ggg₁ 1 have gg₅: (2+(p+1)/2).factorial ∣ p.factorial := by exact factorial_dvd_factorial gg₄ have gg₆: (2:ℕ).factorial * ((p+1)/2).factorial ∣ p.factorial := by refine dvd_trans ?_ gg₅ exact (2:ℕ).factorial_mul_factorial_dvd_factorial_add ((p + 1) / 2) have gg₇: 2 * ((p+1)/2) ∣ p.factorial := by refine dvd_trans ?_ gg₆ simp refine mul_dvd_mul_left 2 ?_ refine Nat.dvd_factorial (by linarith[gg₃]) (by linarith) have gg₈: 2 * ((p+1)/2) * (p + 1) ∣ p.factorial * (p + 1) := by refine mul_dvd_mul_right ?_ (p + 1) exact gg₇ rw [gg₂] at gg₁ exact dvd_trans gg₈ gg₁ lemma imo_2022_p5_12_12 (b p : ℕ) -- (hp : Nat.Prime p) -- (h₁ : p ^ p = b ! + p) (hp5 : 5 ≤ p) -- (hpb : p < b) -- (g₁ : p + 1 ≤ b) -- (g₂ : 2 ∣ p + 1) (gg₁ : (p + 1)! ∣ b !) (gg₂ : (p + 1)! = p ! * (p + 1)) (gg₃ : 3 ≤ (p + 1) / 2) : 2 * ((p + 1) / 2) * (p + 1) ∣ b ! := by have gg₄: 2 + (p+1)/2 ≤ p := by -- strong induction refine Nat.le_induction ?_ ?_ p (hp5) . norm_num . intros n _ h₂ ring_nf have ggg₁: (n / 2).succ ≤ (n + 1) / 2 + 1 := by rw [← succ_eq_add_one] refine Nat.succ_le_succ ?_ refine Nat.div_le_div_right ?_ linarith simp nth_rewrite 1 [← mul_one 2] rw [Nat.two_mul 1, add_assoc] refine Nat.add_le_add_left ?_ 1 refine le_trans ?_ h₂ rw [add_comm 2 _] nth_rewrite 3 [← mul_one 2] rw [Nat.two_mul 1, ← add_assoc, add_comm 1] exact Nat.add_le_add_right ggg₁ 1 have gg₅: (2+(p+1)/2).factorial ∣ p.factorial := by exact factorial_dvd_factorial gg₄ have gg₆: (2:ℕ).factorial * ((p+1)/2).factorial ∣ p.factorial := by refine dvd_trans ?_ gg₅ exact (2:ℕ).factorial_mul_factorial_dvd_factorial_add ((p + 1) / 2) have gg₇: 2 * ((p+1)/2) ∣ p.factorial := by refine dvd_trans ?_ gg₆ simp refine mul_dvd_mul_left 2 ?_ refine Nat.dvd_factorial (by linarith[gg₃]) (by linarith) have gg₈: 2 * ((p+1)/2) * (p + 1) ∣ p.factorial * (p + 1) := by refine mul_dvd_mul_right ?_ (p + 1) exact gg₇ rw [gg₂] at gg₁ exact dvd_trans gg₈ gg₁ lemma imo_2022_p5_12_13 -- (b : ℕ) (p : ℕ) -- (hp : Nat.Prime p) -- (h₁ : p ^ p = b ! + p) (hp5 : 5 ≤ p) : -- (hpb : p < b) -- (g₁ : p + 1 ≤ b) -- (g₂ : 2 ∣ p + 1) -- (gg₁ : (p + 1)! ∣ b !) -- (gg₂ : (p + 1)! = p ! * (p + 1)) -- (gg₃ : 3 ≤ (p + 1) / 2) : 2 + (p + 1) / 2 ≤ p := by refine Nat.le_induction ?_ ?_ p (hp5) . norm_num . intros n _ h₂ ring_nf have ggg₁: (n / 2).succ ≤ (n + 1) / 2 + 1 := by rw [← succ_eq_add_one] refine Nat.succ_le_succ ?_ refine Nat.div_le_div_right ?_ linarith simp nth_rewrite 1 [← mul_one 2] rw [Nat.two_mul 1, add_assoc] refine Nat.add_le_add_left ?_ 1 refine le_trans ?_ h₂ rw [add_comm 2 _] nth_rewrite 3 [← mul_one 2] rw [Nat.two_mul 1, ← add_assoc, add_comm 1] exact Nat.add_le_add_right ggg₁ 1 lemma imo_2022_p5_12_14 : -- (b p : ℕ) -- (hp : Nat.Prime p) -- (h₁ : p ^ p = b ! + p) -- (hp5 : 5 ≤ p) -- (hpb : p < b) -- (g₁ : p + 1 ≤ b) -- (g₂ : 2 ∣ p + 1) -- (gg₁ : (p + 1)! ∣ b !) -- (gg₂ : (p + 1)! = p ! * (p + 1)) -- (gg₃ : 3 ≤ (p + 1) / 2) : ∀ (n : ℕ), 5 ≤ n → 2 + (n + 1) / 2 ≤ n → 2 + (n + 1 + 1) / 2 ≤ n + 1 := by intros n _ h₂ ring_nf have ggg₁: (n / 2).succ ≤ (n + 1) / 2 + 1 := by rw [← succ_eq_add_one] refine Nat.succ_le_succ ?_ refine Nat.div_le_div_right ?_ linarith simp nth_rewrite 1 [← mul_one 2] rw [Nat.two_mul 1, add_assoc] refine Nat.add_le_add_left ?_ 1 refine le_trans ?_ h₂ rw [add_comm 2 _] nth_rewrite 3 [← mul_one 2] rw [Nat.two_mul 1, ← add_assoc, add_comm 1] exact Nat.add_le_add_right ggg₁ 1 lemma imo_2022_p5_12_15 -- (b p : ℕ) -- (hp : Nat.Prime p) -- (h₁ : p ^ p = b ! + p) -- (hp5 : 5 ≤ p) -- (hpb : p < b) -- (g₁ : p + 1 ≤ b) -- (g₂ : 2 ∣ p + 1) -- (gg₁ : (p + 1)! ∣ b !) -- (gg₂ : (p + 1)! = p ! * (p + 1)) -- (gg₃ : 3 ≤ (p + 1) / 2) (n : ℕ) -- (hmn : 5 ≤ n) (h₂ : 2 + (n + 1) / 2 ≤ n) : 2 + (2 + n) / 2 ≤ 1 + n := by ring_nf have ggg₁: (n / 2).succ ≤ (n + 1) / 2 + 1 := by rw [← succ_eq_add_one] refine Nat.succ_le_succ ?_ refine Nat.div_le_div_right ?_ linarith simp nth_rewrite 1 [← mul_one 2] rw [Nat.two_mul 1, add_assoc] refine Nat.add_le_add_left ?_ 1 refine le_trans ?_ h₂ rw [add_comm 2 _] nth_rewrite 3 [← mul_one 2] rw [Nat.two_mul 1, ← add_assoc, add_comm 1] exact Nat.add_le_add_right ggg₁ 1 lemma imo_2022_p5_12_16 -- (b p : ℕ) -- (hp : Nat.Prime p) -- (h₁ : p ^ p = b ! + p) -- (hp5 : 5 ≤ p) -- (hpb : p < b) -- (g₁ : p + 1 ≤ b) -- (g₂ : 2 ∣ p + 1) -- (gg₁ : (p + 1)! ∣ b !) -- (gg₂ : (p + 1)! = p ! * (p + 1)) -- (gg₃ : 3 ≤ (p + 1) / 2) (n : ℕ) : -- (hmn : 5 ≤ n) -- (h₂ : 2 + (n + 1) / 2 ≤ n) : succ (n / 2) ≤ (n + 1) / 2 + 1 := by rw [← succ_eq_add_one] refine Nat.succ_le_succ ?_ refine Nat.div_le_div_right ?_ linarith lemma imo_2022_p5_12_17 -- (b p : ℕ) -- (hp : Nat.Prime p) -- (h₁ : p ^ p = b ! + p) -- (hp5 : 5 ≤ p) -- (hpb : p < b) -- (g₁ : p + 1 ≤ b) -- (g₂ : 2 ∣ p + 1) -- (gg₁ : (p + 1)! ∣ b !) -- (gg₂ : (p + 1)! = p ! * (p + 1)) -- (gg₃ : 3 ≤ (p + 1) / 2) (n : ℕ) -- (hmn : 5 ≤ n) (h₂ : 2 + (n + 1) / 2 ≤ n) (ggg₁ : succ (n / 2) ≤ (n + 1) / 2 + 1) : 2 + succ (n / 2) ≤ 1 + n := by nth_rewrite 1 [← mul_one 2] rw [Nat.two_mul 1, add_assoc] refine Nat.add_le_add_left ?_ 1 refine le_trans ?_ h₂ rw [add_comm 2 _] nth_rewrite 3 [← mul_one 2] rw [Nat.two_mul 1, ← add_assoc, add_comm 1] exact Nat.add_le_add_right ggg₁ 1 lemma imo_2022_p5_12_18 -- (b p : ℕ) -- (hp : Nat.Prime p) -- (h₁ : p ^ p = b ! + p) -- (hp5 : 5 ≤ p) -- (hpb : p < b) -- (g₁ : p + 1 ≤ b) -- (g₂ : 2 ∣ p + 1) -- (gg₁ : (p + 1)! ∣ b !) -- (gg₂ : (p + 1)! = p ! * (p + 1)) -- (gg₃ : 3 ≤ (p + 1) / 2) (n : ℕ) -- (hmn : 5 ≤ n) (h₂ : 2 + (n + 1) / 2 ≤ n) (ggg₁ : succ (n / 2) ≤ (n + 1) / 2 + 1) : 1 + succ (n / 2) ≤ n := by refine le_trans ?_ h₂ rw [add_comm 2 _] nth_rewrite 3 [← mul_one 2] rw [Nat.two_mul 1, ← add_assoc, add_comm 1] exact Nat.add_le_add_right ggg₁ 1 lemma imo_2022_p5_12_19 -- (b p : ℕ) -- (hp : Nat.Prime p) -- (h₁ : p ^ p = b ! + p) -- (hp5 : 5 ≤ p) -- (hpb : p < b) -- (g₁ : p + 1 ≤ b) -- (g₂ : 2 ∣ p + 1) -- (gg₁ : (p + 1)! ∣ b !) -- (gg₂ : (p + 1)! = p ! * (p + 1)) -- (gg₃ : 3 ≤ (p + 1) / 2) (n : ℕ) -- (hmn : 5 ≤ n) (h₂ : 2 + (n + 1) / 2 ≤ n) -- (ggg₁ : succ (n / 2) ≤ (n + 1) / 2 + 1) (g₃ : 1 + succ (n / 2) ≤ (n + 1) / 2 + 2 * 1) : 1 + succ (n / 2) ≤ n := by refine le_trans ?_ h₂ rw [add_comm 2 _] nth_rewrite 3 [← mul_one 2] rw [Nat.two_mul 1, ← add_assoc] exact g₃ lemma imo_2022_p5_12_20 -- (b p : ℕ) -- (hp : Nat.Prime p) -- (h₁ : p ^ p = b ! + p) -- (hp5 : 5 ≤ p) -- (hpb : p < b) -- (g₁ : p + 1 ≤ b) -- (g₂ : 2 ∣ p + 1) -- (gg₁ : (p + 1)! ∣ b !) -- (gg₂ : (p + 1)! = p ! * (p + 1)) -- (gg₃ : 3 ≤ (p + 1) / 2) (n : ℕ) -- (hmn : 5 ≤ n) -- (h₂ : 2 + (n + 1) / 2 ≤ n) (ggg₁ : succ (n / 2) ≤ (n + 1) / 2 + 1) : 1 + succ (n / 2) ≤ (n + 1) / 2 + 2 := by nth_rewrite 3 [← mul_one 2] rw [Nat.two_mul 1, ← add_assoc, add_comm 1] exact Nat.add_le_add_right ggg₁ 1 lemma imo_2022_p5_12_21 (b p : ℕ) -- (hp : Nat.Prime p) -- (h₁ : p ^ p = b ! + p) -- (hp5 : 5 ≤ p) -- (hpb : p < b) -- (g₁ : p + 1 ≤ b) -- (g₂ : 2 ∣ p + 1) (gg₁ : (p + 1)! ∣ b !) (gg₂ : (p + 1)! = p ! * (p + 1)) (gg₃ : 3 ≤ (p + 1) / 2) (gg₄ : 2 + (p + 1) / 2 ≤ p) : 2 * ((p + 1) / 2) * (p + 1) ∣ b ! := by have gg₅: (2+(p+1)/2).factorial ∣ p.factorial := by exact factorial_dvd_factorial gg₄ have gg₆: (2:ℕ).factorial * ((p+1)/2).factorial ∣ p.factorial := by refine dvd_trans ?_ gg₅ exact (2:ℕ).factorial_mul_factorial_dvd_factorial_add ((p + 1) / 2) have gg₇: 2 * ((p+1)/2) ∣ p.factorial := by refine dvd_trans ?_ gg₆ simp refine mul_dvd_mul_left 2 ?_ refine Nat.dvd_factorial (by linarith[gg₃]) (by linarith) have gg₈: 2 * ((p+1)/2) * (p + 1) ∣ p.factorial * (p + 1) := by refine mul_dvd_mul_right ?_ (p + 1) exact gg₇ rw [gg₂] at gg₁ exact dvd_trans gg₈ gg₁ lemma imo_2022_p5_12_22 (b p : ℕ) -- (hp : Nat.Prime p) -- (h₁ : p ^ p = b ! + p) -- (hp5 : 5 ≤ p) -- (hpb : p < b) -- (g₁ : p + 1 ≤ b) -- (g₂ : 2 ∣ p + 1) (gg₁ : (p + 1)! ∣ b !) (gg₂ : (p + 1)! = p ! * (p + 1)) (gg₃ : 3 ≤ (p + 1) / 2) (gg₄ : 2 + (p + 1) / 2 ≤ p) (gg₅ : (2 + (p + 1) / 2)! ∣ p !) : 2 * ((p + 1) / 2) * (p + 1) ∣ b ! := by have gg₆: (2:ℕ).factorial * ((p+1)/2).factorial ∣ p.factorial := by refine dvd_trans ?_ gg₅ exact (2:ℕ).factorial_mul_factorial_dvd_factorial_add ((p + 1) / 2) have gg₇: 2 * ((p+1)/2) ∣ p.factorial := by refine dvd_trans ?_ gg₆ simp refine mul_dvd_mul_left 2 ?_ refine Nat.dvd_factorial (by linarith[gg₃]) (by linarith) have gg₈: 2 * ((p+1)/2) * (p + 1) ∣ p.factorial * (p + 1) := by refine mul_dvd_mul_right ?_ (p + 1) exact gg₇ rw [gg₂] at gg₁ exact dvd_trans gg₈ gg₁ lemma imo_2022_p5_12_23 -- (b : ℕ) (p : ℕ) -- (hp : Nat.Prime p) -- (h₁ : p ^ p = b ! + p) -- (hp5 : 5 ≤ p) -- (hpb : p < b) -- (g₁ : p + 1 ≤ b) -- (g₂ : 2 ∣ p + 1) -- (gg₁ : (p + 1)! ∣ b !) -- (gg₂ : (p + 1)! = p ! * (p + 1)) -- (gg₃ : 3 ≤ (p + 1) / 2) -- (gg₄ : 2 + (p + 1) / 2 ≤ p) (gg₅ : (2 + (p + 1) / 2)! ∣ p !) : 2! * ((p + 1) / 2)! ∣ p ! := by refine dvd_trans ?_ gg₅ exact (2:ℕ).factorial_mul_factorial_dvd_factorial_add ((p + 1) / 2) lemma imo_2022_p5_12_24 (b p : ℕ) -- (hp : Nat.Prime p) -- (h₁ : p ^ p = b ! + p) -- (hp5 : 5 ≤ p) -- (hpb : p < b) -- (g₁ : p + 1 ≤ b) -- (g₂ : 2 ∣ p + 1) (gg₁ : (p + 1)! ∣ b !) (gg₂ : (p + 1)! = p ! * (p + 1)) (gg₃ : 3 ≤ (p + 1) / 2) (gg₄ : 2 + (p + 1) / 2 ≤ p) (gg₅ : (2 + (p + 1) / 2)! ∣ p !) (gg₆ : 2! * ((p + 1) / 2)! ∣ p !) : 2 * ((p + 1) / 2) * (p + 1) ∣ b ! := by have gg₇: 2 * ((p+1)/2) ∣ p.factorial := by refine dvd_trans ?_ gg₆ simp refine mul_dvd_mul_left 2 ?_ refine Nat.dvd_factorial (by linarith[gg₃]) (by linarith) have gg₈: 2 * ((p+1)/2) * (p + 1) ∣ p.factorial * (p + 1) := by refine mul_dvd_mul_right ?_ (p + 1) exact gg₇ rw [gg₂] at gg₁ exact dvd_trans gg₈ gg₁ lemma imo_2022_p5_12_25 -- (b : ℕ) (p : ℕ) -- (hp : Nat.Prime p) -- (h₁ : p ^ p = b ! + p) -- (hp5 : 5 ≤ p) -- (hpb : p < b) -- (g₁ : p + 1 ≤ b) -- (g₂ : 2 ∣ p + 1) -- (gg₁ : (p + 1)! ∣ b !) -- (gg₂ : (p + 1)! = p ! * (p + 1)) (gg₃ : 3 ≤ (p + 1) / 2) -- (gg₄ : 2 + (p + 1) / 2 ≤ p) -- (gg₅ : (2 + (p + 1) / 2)! ∣ p !) (gg₆ : 2! * ((p + 1) / 2)! ∣ p !) : 2 * ((p + 1) / 2) ∣ p ! := by refine dvd_trans ?_ gg₆ simp refine mul_dvd_mul_left 2 ?_ refine Nat.dvd_factorial (by linarith[gg₃]) (by linarith) lemma imo_2022_p5_12_26 -- (b : ℕ) (p : ℕ) -- (hp : Nat.Prime p) -- (h₁ : p ^ p = b ! + p) -- (hp5 : 5 ≤ p) -- (hpb : p < b) -- (g₁ : p + 1 ≤ b) -- (g₂ : 2 ∣ p + 1) -- (gg₁ : (p + 1)! ∣ b !) -- (gg₂ : (p + 1)! = p ! * (p + 1)) (gg₃ : 3 ≤ (p + 1) / 2) : -- (gg₄ : 2 + (p + 1) / 2 ≤ p) -- (gg₅ : (2 + (p + 1) / 2)! ∣ p !) -- (gg₆ : 2! * ((p + 1) / 2)! ∣ p !) : 2 * ((p + 1) / 2) ∣ 2 * ((p + 1) / 2)! := by refine mul_dvd_mul_left 2 ?_ refine Nat.dvd_factorial (by linarith[gg₃]) (by linarith) lemma imo_2022_p5_12_27 (b p : ℕ) -- (hp : Nat.Prime p) -- (h₁ : p ^ p = b ! + p) -- (hp5 : 5 ≤ p) -- (hpb : p < b) -- (g₁ : p + 1 ≤ b) -- (g₂ : 2 ∣ p + 1) (gg₁ : (p + 1)! ∣ b !) (gg₂ : (p + 1)! = p ! * (p + 1)) (gg₃ : 3 ≤ (p + 1) / 2) (gg₄ : 2 + (p + 1) / 2 ≤ p) (gg₅ : (2 + (p + 1) / 2)! ∣ p !) (gg₆ : 2! * ((p + 1) / 2)! ∣ p !) (gg₇ : 2 * ((p + 1) / 2) ∣ p !) : 2 * ((p + 1) / 2) * (p + 1) ∣ b ! := by have gg₈: 2 * ((p+1)/2) * (p + 1) ∣ p.factorial * (p + 1) := by refine mul_dvd_mul_right ?_ (p + 1) exact gg₇ rw [gg₂] at gg₁ exact dvd_trans gg₈ gg₁ lemma imo_2022_p5_12_28 -- (b : ℕ) (p : ℕ) -- (hp : Nat.Prime p) -- (h₁ : p ^ p = b ! + p) -- (hp5 : 5 ≤ p) -- (hpb : p < b) -- (g₁ : p + 1 ≤ b) -- (g₂ : 2 ∣ p + 1) -- (gg₁ : (p + 1)! ∣ b !) -- (gg₂ : (p + 1)! = p ! * (p + 1)) -- (gg₃ : 3 ≤ (p + 1) / 2) -- (gg₄ : 2 + (p + 1) / 2 ≤ p) -- (gg₅ : (2 + (p + 1) / 2)! ∣ p !) -- (gg₆ : 2! * ((p + 1) / 2)! ∣ p !) (gg₇ : 2 * ((p + 1) / 2) ∣ p !) : 2 * ((p + 1) / 2) * (p + 1) ∣ p ! * (p + 1) := by refine mul_dvd_mul_right ?_ (p + 1) exact gg₇ lemma imo_2022_p5_12_29 (b p : ℕ) -- (hp : Nat.Prime p) -- (h₁ : p ^ p = b ! + p) -- (hp5 : 5 ≤ p) -- (hpb : p < b) -- (g₁ : p + 1 ≤ b) -- (g₂ : 2 ∣ p + 1) (gg₁ : (p + 1)! ∣ b !) (gg₂ : (p + 1)! = p ! * (p + 1)) (gg₃ : 3 ≤ (p + 1) / 2) (gg₄ : 2 + (p + 1) / 2 ≤ p) (gg₅ : (2 + (p + 1) / 2)! ∣ p !) (gg₆ : 2! * ((p + 1) / 2)! ∣ p !) (gg₇ : 2 * ((p + 1) / 2) ∣ p !) (gg₈ : 2 * ((p + 1) / 2) * (p + 1) ∣ p ! * (p + 1)) : 2 * ((p + 1) / 2) * (p + 1) ∣ b ! := by rw [gg₂] at gg₁ exact dvd_trans gg₈ gg₁ lemma imo_2022_p5_12_30 (b p : ℕ) -- (hp : Nat.Prime p) (h₁ : p ^ p = b ! + p) -- (hp5 : 5 ≤ p) -- (hpb : p < b) (g₁ : p + 1 ≤ b) (g₂ : 2 ∣ p + 1) (g₃ : 2 * ((p + 1) / 2) * (p + 1) ∣ b !) : (p + 1) ^ 2 ∣ b ! := by have g₄: 2 * ((p+1)/2) = (p + 1) := by exact Nat.mul_div_cancel' g₂ rw [g₄] at g₃ ring_nf at * exact g₃ lemma imo_2022_p5_12_31 (b p : ℕ) -- (hp : Nat.Prime p) (h₁ : p ^ p = b ! + p) -- (hp5 : 5 ≤ p) -- (hpb : p < b) (g₁ : p + 1 ≤ b) (g₂ : 2 ∣ p + 1) (g₃ : 2 * ((p + 1) / 2) * (p + 1) ∣ b !) (g₄ : 2 * ((p + 1) / 2) = p + 1) : (p + 1) ^ 2 ∣ b ! := by rw [g₄] at g₃ ring_nf at * exact g₃ lemma imo_2022_p5_12_32 (b p : ℕ) (hp : Nat.Prime p) (h₁ : p ^ p = b ! + p) (hp5 : 5 ≤ p) -- (hpb : p < b) (h₂ : (p + 1) ^ 2 ∣ b !) : False := by have h₃: b.factorial = p ^ p - p := by exact eq_tsub_of_add_eq (h₁.symm) rw [h₃] at h₂ exact imo_2022_p5_10 p hp hp5 h₂ lemma imo_2022_p5_13 (a b p: ℕ) (hp: Nat.Prime p) (h₂: p ∣ a) (hb2p: 2 * p ≤ b) : (p ^ 2 ∣ a ^ p - b.factorial) := by have g₁: p^p ∣ a^p := by exact pow_dvd_pow_of_dvd h₂ p have g₂: 2 ≤ p := by exact Prime.two_le hp have h₃: p^2 ∣ a^p := by exact pow_dvd_of_le_of_pow_dvd g₂ g₁ have g₃: (2*p).factorial ∣ b.factorial := by exact factorial_dvd_factorial hb2p have g₄: p.factorial * p.factorial ∣ (p+p).factorial := by exact factorial_mul_factorial_dvd_factorial_add p p rw [← pow_two, ← two_mul] at g₄ have g₅: p ∣ p.factorial := by exact Nat.dvd_factorial (by linarith) (by linarith) have h₄: p ^ 2 ∣ p.factorial ^ 2 := by exact pow_dvd_pow_of_dvd g₅ 2 have g₆: p ^ 2 ∣ (2 * p).factorial := by exact dvd_trans h₄ g₄ have h₅: p^2 ∣ b.factorial := by exact dvd_trans g₆ g₃ exact dvd_sub' h₃ h₅ lemma imo_2022_p5_13_1 (a b p : ℕ) (hp : Nat.Prime p) -- (h₂ : p ∣ a) (hb2p : 2 * p ≤ b) (g₁ : p ^ p ∣ a ^ p) : p ^ 2 ∣ a ^ p - b ! := by have g₂: 2 ≤ p := by exact Prime.two_le hp have h₃: p^2 ∣ a^p := by exact pow_dvd_of_le_of_pow_dvd g₂ g₁ have g₃: (2*p).factorial ∣ b.factorial := by exact factorial_dvd_factorial hb2p have g₄: p.factorial * p.factorial ∣ (p+p).factorial := by exact factorial_mul_factorial_dvd_factorial_add p p rw [← pow_two, ← two_mul] at g₄ have g₅: p ∣ p.factorial := by exact Nat.dvd_factorial (by linarith) (by linarith) have h₄: p ^ 2 ∣ p.factorial ^ 2 := by exact pow_dvd_pow_of_dvd g₅ 2 have g₆: p ^ 2 ∣ (2 * p).factorial := by exact dvd_trans h₄ g₄ have h₅: p^2 ∣ b.factorial := by exact dvd_trans g₆ g₃ exact dvd_sub' h₃ h₅ lemma imo_2022_p5_13_2 (a b p : ℕ) -- (hp : Nat.Prime p) -- (h₂ : p ∣ a) (hb2p : 2 * p ≤ b) (g₁ : p ^ p ∣ a ^ p) (g₂ : 2 ≤ p) : p ^ 2 ∣ a ^ p - b ! := by have h₃: p^2 ∣ a^p := by exact pow_dvd_of_le_of_pow_dvd g₂ g₁ have g₃: (2*p).factorial ∣ b.factorial := by exact factorial_dvd_factorial hb2p have g₄: p.factorial * p.factorial ∣ (p+p).factorial := by exact factorial_mul_factorial_dvd_factorial_add p p rw [← pow_two, ← two_mul] at g₄ have g₅: p ∣ p.factorial := by exact Nat.dvd_factorial (by linarith) (by linarith) have h₄: p ^ 2 ∣ p.factorial ^ 2 := by exact pow_dvd_pow_of_dvd g₅ 2 have g₆: p ^ 2 ∣ (2 * p).factorial := by exact dvd_trans h₄ g₄ have h₅: p^2 ∣ b.factorial := by exact dvd_trans g₆ g₃ exact dvd_sub' h₃ h₅ lemma imo_2022_p5_13_3 (a b p : ℕ) -- (hp : Nat.Prime p) -- (h₂ : p ∣ a) (hb2p : 2 * p ≤ b) -- (g₁ : p ^ p ∣ a ^ p) (g₂ : 2 ≤ p) (h₃ : p ^ 2 ∣ a ^ p) : p ^ 2 ∣ a ^ p - b ! := by have g₃: (2*p).factorial ∣ b.factorial := by exact factorial_dvd_factorial hb2p have g₄: p.factorial * p.factorial ∣ (p+p).factorial := by exact factorial_mul_factorial_dvd_factorial_add p p rw [← pow_two, ← two_mul] at g₄ have g₅: p ∣ p.factorial := by exact Nat.dvd_factorial (by linarith) (by linarith) have h₄: p ^ 2 ∣ p.factorial ^ 2 := by exact pow_dvd_pow_of_dvd g₅ 2 have g₆: p ^ 2 ∣ (2 * p).factorial := by exact dvd_trans h₄ g₄ have h₅: p^2 ∣ b.factorial := by exact dvd_trans g₆ g₃ exact dvd_sub' h₃ h₅ lemma imo_2022_p5_13_4 (a p : ℕ) -- (b : ℕ) (hp : Nat.Prime p) (h₂ : p ∣ a) : -- (hb2p : 2 * p ≤ b) : p ^ 2 ∣ a ^ p := by have g₁: p^p ∣ a^p := by exact pow_dvd_pow_of_dvd h₂ p have g₂: 2 ≤ p := by exact Prime.two_le hp exact pow_dvd_of_le_of_pow_dvd g₂ g₁ lemma imo_2022_p5_13_5 (a b p : ℕ) -- (hp : Nat.Prime p) -- (h₂ : p ∣ a) -- (hb2p : 2 * p ≤ b) -- (g₁ : p ^ p ∣ a ^ p) (g₂ : 2 ≤ p) (h₃ : p ^ 2 ∣ a ^ p) (g₃ : (2 * p)! ∣ b !) : p ^ 2 ∣ a ^ p - b ! := by have g₄: p.factorial * p.factorial ∣ (p+p).factorial := by exact factorial_mul_factorial_dvd_factorial_add p p rw [← pow_two, ← two_mul] at g₄ have g₅: p ∣ p.factorial := by exact Nat.dvd_factorial (by linarith) (by linarith) have h₄: p ^ 2 ∣ p.factorial ^ 2 := by exact pow_dvd_pow_of_dvd g₅ 2 have g₆: p ^ 2 ∣ (2 * p).factorial := by exact dvd_trans h₄ g₄ have h₅: p^2 ∣ b.factorial := by exact dvd_trans g₆ g₃ exact dvd_sub' h₃ h₅ lemma imo_2022_p5_13_6 (a b p : ℕ) -- (hp : Nat.Prime p) -- (h₂ : p ∣ a) -- (hb2p : 2 * p ≤ b) -- (g₁ : p ^ p ∣ a ^ p) (g₂ : 2 ≤ p) (h₃ : p ^ 2 ∣ a ^ p) (g₃ : (2 * p)! ∣ b !) (g₄ : p ! * p ! ∣ (p + p)!) : p ^ 2 ∣ a ^ p - b ! := by rw [← pow_two, ← two_mul] at g₄ have g₅: p ∣ p.factorial := by exact Nat.dvd_factorial (by linarith) (by linarith) have h₄: p ^ 2 ∣ p.factorial ^ 2 := by exact pow_dvd_pow_of_dvd g₅ 2 have g₆: p ^ 2 ∣ (2 * p).factorial := by exact dvd_trans h₄ g₄ have h₅: p^2 ∣ b.factorial := by exact dvd_trans g₆ g₃ exact dvd_sub' h₃ h₅ lemma imo_2022_p5_13_7 (a b p : ℕ) -- (hp : Nat.Prime p) -- (h₂ : p ∣ a) -- (hb2p : 2 * p ≤ b) -- (g₁ : p ^ p ∣ a ^ p) -- (g₂ : 2 ≤ p) (h₃ : p ^ 2 ∣ a ^ p) (g₃ : (2 * p)! ∣ b !) (g₄ : p ! ^ 2 ∣ (2 * p)!) (g₅ : p ∣ p !) : p ^ 2 ∣ a ^ p - b ! := by have h₄: p ^ 2 ∣ p.factorial ^ 2 := by exact pow_dvd_pow_of_dvd g₅ 2 have g₆: p ^ 2 ∣ (2 * p).factorial := by exact dvd_trans h₄ g₄ have h₅: p^2 ∣ b.factorial := by exact dvd_trans g₆ g₃ exact dvd_sub' h₃ h₅ lemma imo_2022_p5_13_8 -- (a b : ℕ) (p : ℕ) -- (hp : Nat.Prime p) -- (h₂ : p ∣ a) -- (hb2p : 2 * p ≤ b) -- (g₁ : p ^ p ∣ a ^ p) (g₂ : 2 ≤ p) : -- (h₃ : p ^ 2 ∣ a ^ p) -- (g₃ : (2 * p)! ∣ b !) -- (g₄ : p ! ^ 2 ∣ (2 * p)!) : p ^ 2 ∣ p ! ^ 2 := by have g₅: p ∣ p.factorial := by exact Nat.dvd_factorial (by linarith) (by linarith) exact pow_dvd_pow_of_dvd g₅ 2 lemma imo_2022_p5_13_9 -- (a b : ℕ) (p : ℕ) -- (hp : Nat.Prime p) -- (h₂ : p ∣ a) -- (hb2p : 2 * p ≤ b) -- (g₁ : p ^ p ∣ a ^ p) (g₂ : 2 ≤ p) : -- (h₃ : p ^ 2 ∣ a ^ p) : -- (g₃ : (2 * p)! ∣ b !) -- (g₄ : p ! ^ 2 ∣ (2 * p)!) p ^ 2 ∣ p ! ^ 2 := by refine pow_dvd_pow_of_dvd ?_ 2 exact Nat.dvd_factorial (by linarith) (by linarith) lemma imo_2022_p5_13_10 (a b p : ℕ) -- (hp : Nat.Prime p) -- (h₂ : p ∣ a) -- (hb2p : 2 * p ≤ b) -- (g₁ : p ^ p ∣ a ^ p) -- (g₂ : 2 ≤ p) (h₃ : p ^ 2 ∣ a ^ p) (g₃ : (2 * p)! ∣ b !) (g₄ : p ! ^ 2 ∣ (2 * p)!) -- (g₅ : p ∣ p !) (h₄ : p ^ 2 ∣ p ! ^ 2) : p ^ 2 ∣ a ^ p - b ! := by have g₆: p ^ 2 ∣ (2 * p).factorial := by exact dvd_trans h₄ g₄ have h₅: p^2 ∣ b.factorial := by exact dvd_trans g₆ g₃ exact dvd_sub' h₃ h₅ lemma imo_2022_p5_13_11 -- (a b : ℕ) (p : ℕ) -- (hp : Nat.Prime p) -- (h₂ : p ∣ a) -- (hb2p : 2 * p ≤ b) -- (g₁ : p ^ p ∣ a ^ p) -- (g₂ : 2 ≤ p) -- (h₃ : p ^ 2 ∣ a ^ p) -- (g₃ : (2 * p)! ∣ b !) (g₅ : p ∣ p !) : p ^ 2 ∣ (2 * p)! := by have h₄: p ^ 2 ∣ p.factorial ^ 2 := by exact pow_dvd_pow_of_dvd g₅ 2 refine dvd_trans h₄ ?_ have g₄: p.factorial * p.factorial ∣ (p+p).factorial := by exact factorial_mul_factorial_dvd_factorial_add p p rw [← pow_two, ← two_mul] at g₄ exact g₄ lemma imo_2022_p5_13_12 (a b p : ℕ) -- (hp : Nat.Prime p) -- (h₂ : p ∣ a) -- (hb2p : 2 * p ≤ b) -- (g₁ : p ^ p ∣ a ^ p) -- (g₂ : 2 ≤ p) (h₃ : p ^ 2 ∣ a ^ p) (g₃ : (2 * p)! ∣ b !) -- (g₄ : p ! ^ 2 ∣ (2 * p)!) -- (g₅ : p ∣ p !) -- (h₄ : p ^ 2 ∣ p ! ^ 2) (g₆ : p ^ 2 ∣ (2 * p)!) : p ^ 2 ∣ a ^ p - b ! := by have h₅: p^2 ∣ b.factorial := by exact dvd_trans g₆ g₃ exact dvd_sub' h₃ h₅ lemma imo_2022_p5_13_13 -- (a : ℕ) (b p : ℕ) -- (hp : Nat.Prime p) -- (h₂ : p ∣ a) -- (hb2p : 2 * p ≤ b) -- (g₁ : p ^ p ∣ a ^ p) -- (g₂ : 2 ≤ p) -- (h₃ : p ^ 2 ∣ a ^ p) (g₃ : (2 * p)! ∣ b !) (g₄ : p ! ^ 2 ∣ (2 * p)!) -- (g₅ : p ∣ p !) (h₄ : p ^ 2 ∣ p ! ^ 2) : p ^ 2 ∣ b ! := by refine dvd_trans ?_ g₃ exact dvd_trans h₄ g₄ lemma imo_2022_p5_13_14 -- (a : ℕ) (b p : ℕ) -- (hp : Nat.Prime p) -- (h₂ : p ∣ a) (hb2p : 2 * p ≤ b) -- (g₁ : p ^ p ∣ a ^ p) -- (g₂ : 2 ≤ p) -- (h₃ : p ^ 2 ∣ a ^ p) (g₄ : p ! ^ 2 ∣ (2 * p)!) -- (g₅ : p ∣ p !) (h₄ : p ^ 2 ∣ p ! ^ 2) : p ^ 2 ∣ b ! := by have g₃: (2*p).factorial ∣ b.factorial := by exact factorial_dvd_factorial hb2p refine dvd_trans ?_ g₃ exact dvd_trans h₄ g₄ lemma imo_2022_p5_14 (a b p : ℕ) (h₀ : 0 < a ∧ 0 < b) (hp : Nat.Prime p) (h₁ : a ^ p = b ! + p) (hbp : b < p) : (a, b, p) = (2, 2, 2) ∨ (a, b, p) = (3, 4, 3) := by exfalso by_cases hab: a ≤ b . have h₂: a ∣ b.factorial := by exact Nat.dvd_factorial h₀.1 hab have g₃: a ∣ b.factorial + p := by rw [← h₁] refine dvd_pow_self a ?_ exact Nat.Prime.ne_zero hp have h₃: a ∣ p := by exact (Nat.dvd_add_right h₂).mp g₃ have h₄: a = 1 := by have g₄: a = 1 ∨ a = p := by exact (Nat.dvd_prime hp).mp h₃ cases' g₄ with g₄₀ g₄₁ . exact g₄₀ . exfalso rw [← g₄₁] at hbp linarith[hbp,hab] rw [h₄] at h₁ simp at h₁ have h₅: 2 ≤ p := by exact Nat.Prime.two_le hp have g₆: 0 < b.factorial := by exact Nat.factorial_pos b have h₇: 1+2 ≤ b.factorial + p := by exact Nat.add_le_add g₆ h₅ rw [← h₁] at h₇ linarith . push_neg at hab have h₂: (b+1)^p ≤ a^p := by refine (Nat.pow_le_pow_iff_left ?_).mpr hab exact Nat.Prime.ne_zero hp have h₃: b^p + p*b + 1 ≤ (b+1)^p := by ring_nf rw [add_assoc] exact imo_2022_p5_1 b p h₀.2 hbp have g₄: p * 1 ≤ p * b := by refine mul_le_mul ?_ ?_ ?_ ?_ . exact rfl.ge . exact h₀.2 . norm_num . exact Nat.zero_le p have g₄: b.factorial ≤ b^b := by exact Nat.factorial_le_pow b have g₅: b^b ≤ b^p := by refine Nat.pow_le_pow_of_le_right h₀.2 ?_ exact le_of_lt hbp linarith lemma imo_2022_p5_14_1 (a b p : ℕ) (h₀ : 0 < a ∧ 0 < b) (hp : Nat.Prime p) (h₁ : a ^ p = b ! + p) (hbp : b < p) (hab : a ≤ b) : False := by have h₂: a ∣ b.factorial := by exact Nat.dvd_factorial h₀.1 hab have g₃: a ∣ b.factorial + p := by rw [← h₁] refine dvd_pow_self a ?_ exact Nat.Prime.ne_zero hp have h₃: a ∣ p := by exact (Nat.dvd_add_right h₂).mp g₃ have h₄: a = 1 := by have g₄: a = 1 ∨ a = p := by exact (Nat.dvd_prime hp).mp h₃ cases' g₄ with g₄₀ g₄₁ . exact g₄₀ . exfalso rw [← g₄₁] at hbp linarith[hbp,hab] rw [h₄] at h₁ simp at h₁ have h₅: 2 ≤ p := by exact Nat.Prime.two_le hp have g₆: 0 < b.factorial := by exact Nat.factorial_pos b have h₇: 1+2 ≤ b.factorial + p := by exact Nat.add_le_add g₆ h₅ rw [← h₁] at h₇ linarith lemma imo_2022_p5_14_2 (a b p : ℕ) (h₀ : 0 < a ∧ 0 < b) (hp : Nat.Prime p) (h₁ : a ^ p = b ! + p) (hbp : b < p) (hab : a ≤ b) (h₂ : a ∣ b !) : False := by have g₃: a ∣ b.factorial + p := by rw [← h₁] refine dvd_pow_self a ?_ exact Nat.Prime.ne_zero hp have h₃: a ∣ p := by exact (Nat.dvd_add_right h₂).mp g₃ have h₄: a = 1 := by have g₄: a = 1 ∨ a = p := by exact (Nat.dvd_prime hp).mp h₃ cases' g₄ with g₄₀ g₄₁ . exact g₄₀ . exfalso rw [← g₄₁] at hbp linarith[hbp,hab] rw [h₄] at h₁ simp at h₁ have h₅: 2 ≤ p := by exact Nat.Prime.two_le hp have g₆: 0 < b.factorial := by exact Nat.factorial_pos b have h₇: 1+2 ≤ b.factorial + p := by exact Nat.add_le_add g₆ h₅ rw [← h₁] at h₇ linarith lemma imo_2022_p5_14_3 (a b p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) (hp : Nat.Prime p) (h₁ : a ^ p = b ! + p) : -- (hbp : b < p) -- (hab : a ≤ b) -- (h₂ : a ∣ b !) : a ∣ b ! + p := by rw [← h₁] refine dvd_pow_self a ?_ exact Nat.Prime.ne_zero hp lemma imo_2022_p5_14_4 (a b p : ℕ) (h₀ : 0 < a ∧ 0 < b) (hp : Nat.Prime p) (h₁ : a ^ p = b ! + p) (hbp : b < p) (hab : a ≤ b) (h₂ : a ∣ b !) (g₃ : a ∣ b ! + p) : False := by have h₃: a ∣ p := by exact (Nat.dvd_add_right h₂).mp g₃ have h₄: a = 1 := by have g₄: a = 1 ∨ a = p := by exact (Nat.dvd_prime hp).mp h₃ cases' g₄ with g₄₀ g₄₁ . exact g₄₀ . exfalso rw [← g₄₁] at hbp linarith[hbp,hab] rw [h₄] at h₁ simp at h₁ have h₅: 2 ≤ p := by exact Nat.Prime.two_le hp have g₆: 0 < b.factorial := by exact Nat.factorial_pos b have h₇: 1 + 2 ≤ b.factorial + p := by exact Nat.add_le_add g₆ h₅ rw [← h₁] at h₇ linarith lemma imo_2022_p5_14_5 (a b p : ℕ) (h₀ : 0 < a ∧ 0 < b) (hp : Nat.Prime p) (h₁ : a ^ p = b ! + p) (hbp : b < p) (hab : a ≤ b) (h₂ : a ∣ b !) (g₃ : a ∣ b ! + p) (h₃ : a ∣ p) : False := by have h₄: a = 1 := by have g₄: a = 1 ∨ a = p := by exact (Nat.dvd_prime hp).mp h₃ cases' g₄ with g₄₀ g₄₁ . exact g₄₀ . exfalso rw [← g₄₁] at hbp linarith[hbp,hab] rw [h₄] at h₁ simp at h₁ have h₅: 2 ≤ p := by exact Nat.Prime.two_le hp have g₆: 0 < b.factorial := by exact Nat.factorial_pos b have h₇: 1+2 ≤ b.factorial + p := by exact Nat.add_le_add g₆ h₅ rw [← h₁] at h₇ linarith lemma imo_2022_p5_14_6 (a b p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) (hp : Nat.Prime p) -- (h₁ : a ^ p = b ! + p) (hbp : b < p) (hab : a ≤ b) (h₂ : a ∣ b !) (g₃ : a ∣ b ! + p) (h₃ : a ∣ p) : a = 1 := by have g₄: a = 1 ∨ a = p := by exact (Nat.dvd_prime hp).mp h₃ cases' g₄ with g₄₀ g₄₁ . exact g₄₀ . exfalso rw [← g₄₁] at hbp linarith[hbp,hab] lemma imo_2022_p5_14_7 (a b p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) -- (hp : Nat.Prime p) -- (h₁ : a ^ p = b ! + p) (hbp : b < p) (hab : a ≤ b) (h₂ : a ∣ b !) (g₃ : a ∣ b ! + p) (h₃ : a ∣ p) (g₄ : a = 1 ∨ a = p) : a = 1 := by cases' g₄ with g₄₀ g₄₁ . exact g₄₀ . exfalso rw [← g₄₁] at hbp linarith[hbp,hab] lemma imo_2022_p5_14_8 (a b p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) -- (hp : Nat.Prime p) -- (h₁ : a ^ p = b ! + p) (hbp : b < p) (hab : a ≤ b) (h₂ : a ∣ b !) (g₃ : a ∣ b ! + p) (h₃ : a ∣ p) (g₄₁ : a = p) : a = 1 := by exfalso rw [← g₄₁] at hbp linarith[hbp,hab] lemma imo_2022_p5_14_9 (a b p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) -- (hp : Nat.Prime p) -- (h₁ : a ^ p = b ! + p) (hbp : b < p) (hab : a ≤ b) (h₂ : a ∣ b !) (g₃ : a ∣ b ! + p) (h₃ : a ∣ p) (g₄₁ : a = p) : False := by rw [← g₄₁] at hbp linarith[hbp,hab] lemma imo_2022_p5_14_10 (a b p : ℕ) (h₀ : 0 < a ∧ 0 < b) (hp : Nat.Prime p) (h₁ : a ^ p = b ! + p) (hbp : b < p) (hab : a ≤ b) (h₂ : a ∣ b !) (g₃ : a ∣ b ! + p) (h₃ : a ∣ p) (h₄ : a = 1) : False := by rw [h₄] at h₁ simp at h₁ have h₅: 2 ≤ p := by exact Nat.Prime.two_le hp have g₆: 0 < b.factorial := by exact Nat.factorial_pos b have h₇: 1+2 ≤ b.factorial + p := by exact Nat.add_le_add g₆ h₅ rw [← h₁] at h₇ linarith lemma imo_2022_p5_14_11 (a b p : ℕ) (h₀ : 0 < a ∧ 0 < b) (hp : Nat.Prime p) (hbp : b < p) -- (hab : a ≤ b) -- (h₂ : a ∣ b !) -- (g₃ : a ∣ b ! + p) -- (h₃ : a ∣ p) -- (h₄ : a = 1) (h₁ : 1 = b ! + p) : False := by have h₅: 2 ≤ p := by exact Nat.Prime.two_le hp have g₆: 0 < b.factorial := by exact Nat.factorial_pos b have h₇: 1+2 ≤ b.factorial + p := by exact Nat.add_le_add g₆ h₅ rw [← h₁] at h₇ linarith lemma imo_2022_p5_14_12 (a b p : ℕ) (h₀ : 0 < a ∧ 0 < b) -- (hp : Nat.Prime p) (hbp : b < p) -- (hab : a ≤ b) -- (h₂ : a ∣ b !) -- (g₃ : a ∣ b ! + p) -- (h₃ : a ∣ p) -- (h₄ : a = 1) (h₁ : 1 = b ! + p) (h₅ : 2 ≤ p) : False := by have g₆: 0 < b.factorial := by exact Nat.factorial_pos b have h₇: 1+2 ≤ b.factorial + p := by exact Nat.add_le_add g₆ h₅ rw [← h₁] at h₇ linarith lemma imo_2022_p5_14_13 (a b p : ℕ) (h₀ : 0 < a ∧ 0 < b) -- (hp : Nat.Prime p) (hbp : b < p) -- (hab : a ≤ b) -- (h₂ : a ∣ b !) -- (g₃ : a ∣ b ! + p) -- (h₃ : a ∣ p) -- (h₄ : a = 1) (h₁ : 1 = b ! + p) (h₅ : 2 ≤ p) (g₆ : 0 < b !) : False := by have h₇: 1+2 ≤ b.factorial + p := by exact Nat.add_le_add g₆ h₅ rw [← h₁] at h₇ linarith lemma imo_2022_p5_14_14 (a b p : ℕ) (h₀ : 0 < a ∧ 0 < b) -- (hp : Nat.Prime p) (hbp : b < p) -- (hab : a ≤ b) -- (h₂ : a ∣ b !) -- (g₃ : a ∣ b ! + p) -- (h₃ : a ∣ p) -- (h₄ : a = 1) (h₁ : 1 = b ! + p) : -- (h₅ : 2 ≤ p) : 1 ≤ b ! := by have g₆: 0 < b.factorial := by exact Nat.factorial_pos b linarith [g₆] lemma imo_2022_p5_14_15 (a b p : ℕ) (h₀ : 0 < a ∧ 0 < b) -- (hp : Nat.Prime p) (hbp : b < p) -- (hab : a ≤ b) -- (h₂ : a ∣ b !) -- (g₃ : a ∣ b ! + p) -- (h₃ : a ∣ p) -- (h₄ : a = 1) (h₁ : 1 = b ! + p) (h₅ : 2 ≤ p) (g₆ : 0 < b !) : -- (h₆ : 1 ≤ b !) : False := by have h₇: 1+2 ≤ b.factorial + p := by exact Nat.add_le_add g₆ h₅ rw [← h₁] at h₇ linarith lemma imo_2022_p5_14_16 -- (a : ℕ) (b p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) (hp : Nat.Prime p) : -- (hbp : b < p) -- (hab : a ≤ b) -- (h₂ : a ∣ b !) -- (g₃ : a ∣ b ! + p) -- (h₃ : a ∣ p) -- (h₄ : a = 1) -- (h₁ : 1 = b ! + p) -- (h₆ : 1 ≤ b !) : 1 + 2 ≤ b ! + p := by have h₅: 2 ≤ p := by exact Nat.Prime.two_le hp have g₆: 0 < b.factorial := by exact Nat.factorial_pos b exact Nat.add_le_add g₆ h₅ lemma imo_2022_p5_14_17 (a b p : ℕ) (h₀ : 0 < a ∧ 0 < b) -- (hp : Nat.Prime p) (hbp : b < p) -- (hab : a ≤ b) -- (h₂ : a ∣ b !) -- (g₃ : a ∣ b ! + p) -- (h₃ : a ∣ p) -- (h₄ : a = 1) (h₁ : 1 = b ! + p) -- (h₅ : 2 ≤ p) -- (g₆ : 0 < b !) -- (h₆ : 1 ≤ b !) (h₇ : 1 + 2 ≤ b ! + p) : False := by rw [← h₁] at h₇ linarith lemma imo_2022_p5_14_18 (a b p : ℕ) (h₀ : 0 < a ∧ 0 < b) (hp : Nat.Prime p) (h₁ : a ^ p = b ! + p) (hbp : b < p) (hab : b < a) : False := by have h₂: (b+1)^p ≤ a^p := by refine (Nat.pow_le_pow_iff_left ?_).mpr hab exact Nat.Prime.ne_zero hp have h₃: b^p + p*b + 1 ≤ (b+1)^p := by ring_nf rw [add_assoc] exact imo_2022_p5_1 b p h₀.2 hbp have g₄: p * 1 ≤ p * b := by refine mul_le_mul ?_ ?_ ?_ ?_ . exact rfl.ge . exact h₀.2 . norm_num . exact Nat.zero_le p have g₄: b.factorial ≤ b^b := by exact Nat.factorial_le_pow b have g₅: b^b ≤ b^p := by refine Nat.pow_le_pow_of_le_right h₀.2 ?_ exact le_of_lt hbp linarith lemma imo_2022_p5_14_19 (a b p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) (hp : Nat.Prime p) -- (h₁ : a ^ p = b ! + p) -- (hbp : b < p) (hab : b < a) : (b + 1) ^ p ≤ a ^ p := by refine (Nat.pow_le_pow_iff_left ?_).mpr hab exact Nat.Prime.ne_zero hp lemma imo_2022_p5_14_20 (a b p : ℕ) (h₀ : 0 < a ∧ 0 < b) -- (hp : Nat.Prime p) (h₁ : a ^ p = b ! + p) (hbp : b < p) -- (hab : b < a) (h₂ : (b + 1) ^ p ≤ a ^ p) : False := by have h₃: b^p + p*b + 1 ≤ (b+1)^p := by ring_nf rw [add_assoc] exact imo_2022_p5_1 b p h₀.2 hbp have g₄: p * 1 ≤ p * b := by refine mul_le_mul ?_ ?_ ?_ ?_ . exact rfl.ge . exact h₀.2 . norm_num . exact Nat.zero_le p have g₄: b.factorial ≤ b^b := by exact Nat.factorial_le_pow b have g₅: b^b ≤ b^p := by refine Nat.pow_le_pow_of_le_right h₀.2 ?_ exact le_of_lt hbp linarith lemma imo_2022_p5_14_21 (a b p : ℕ) (h₀ : 0 < a ∧ 0 < b) -- (hp : Nat.Prime p) -- (h₁ : a ^ p = b ! + p) (hbp : b < p) : -- (hab : b < a) -- (h₂ : (b + 1) ^ p ≤ a ^ p) : b ^ p + p * b + 1 ≤ (b + 1) ^ p := by ring_nf rw [add_assoc] exact imo_2022_p5_1 b p h₀.2 hbp lemma imo_2022_p5_14_22 (a b p : ℕ) (h₀ : 0 < a ∧ 0 < b) -- (hp : Nat.Prime p) (h₁ : a ^ p = b ! + p) (hbp : b < p) -- (hab : b < a) (h₂ : (b + 1) ^ p ≤ a ^ p) (h₃ : b ^ p + p * b + 1 ≤ (b + 1) ^ p) : False := by have g₄: p * 1 ≤ p * b := by refine mul_le_mul ?_ ?_ ?_ ?_ . exact rfl.ge . exact h₀.2 . norm_num . exact Nat.zero_le p have g₄: b.factorial ≤ b^b := by exact Nat.factorial_le_pow b have g₅: b^b ≤ b^p := by refine Nat.pow_le_pow_of_le_right h₀.2 ?_ exact le_of_lt hbp linarith lemma imo_2022_p5_14_23 (a b p : ℕ) (h₀ : 0 < a ∧ 0 < b) : -- (hp : Nat.Prime p) -- (h₁ : a ^ p = b ! + p) -- (hbp : b < p) -- (hab : b < a) -- (h₂ : (b + 1) ^ p ≤ a ^ p) -- (h₃ : b ^ p + p * b + 1 ≤ (b + 1) ^ p) : p * 1 ≤ p * b := by refine mul_le_mul ?_ ?_ ?_ ?_ . exact rfl.ge . exact h₀.2 . norm_num . exact Nat.zero_le p lemma imo_2022_p5_14_24 (a b p : ℕ) (h₀ : 0 < a ∧ 0 < b) -- (hp : Nat.Prime p) (h₁ : a ^ p = b ! + p) (hbp : b < p) -- (hab : b < a) (h₂ : (b + 1) ^ p ≤ a ^ p) (h₃ : b ^ p + p * b + 1 ≤ (b + 1) ^ p) (g₄ : p * 1 ≤ p * b) : False := by have g₄: b.factorial ≤ b^b := by exact Nat.factorial_le_pow b have g₅: b^b ≤ b^p := by refine Nat.pow_le_pow_of_le_right h₀.2 ?_ exact le_of_lt hbp linarith lemma imo_2022_p5_14_25 (a b p : ℕ) (h₀ : 0 < a ∧ 0 < b) -- (hp : Nat.Prime p) (h₁ : a ^ p = b ! + p) (hbp : b < p) -- (hab : b < a) (h₂ : (b + 1) ^ p ≤ a ^ p) (h₃ : b ^ p + p * b + 1 ≤ (b + 1) ^ p) -- (g₄ : p * 1 ≤ p * b) (h₄ : b ^ p + p < b ^ p + p * b + 1) : False := by have g₄: b.factorial ≤ b^b := by exact Nat.factorial_le_pow b have g₅: b^b ≤ b^p := by refine Nat.pow_le_pow_of_le_right h₀.2 ?_ exact le_of_lt hbp linarith lemma imo_2022_p5_14_26 (a b p : ℕ) (h₀ : 0 < a ∧ 0 < b) -- (hp : Nat.Prime p) (h₁ : a ^ p = b ! + p) (hbp : b < p) -- (hab : b < a) (h₂ : (b + 1) ^ p ≤ a ^ p) (h₃ : b ^ p + p * b + 1 ≤ (b + 1) ^ p) -- (g4 : p * 1 ≤ p * b) (h₄ : b ^ p + p < b ^ p + p * b + 1) (g₄ : b ! ≤ b ^ b) : False := by have g₅: b^b ≤ b^p := by refine Nat.pow_le_pow_of_le_right h₀.2 ?_ exact le_of_lt hbp linarith lemma imo_2022_p5_14_27 (a b p : ℕ) (h₀ : 0 < a ∧ 0 < b) -- (hp : Nat.Prime p) -- (h₁ : a ^ p = b ! + p) (hbp : b < p) : -- (hab : b < a) -- (h₂ : (b + 1) ^ p ≤ a ^ p) -- (h₃ : b ^ p + p * b + 1 ≤ (b + 1) ^ p) -- (g4 : p * 1 ≤ p * b) -- (h₄ : b ^ p + p < b ^ p + p * b + 1) -- (g₄ : b ! ≤ b ^ b) : b ^ b ≤ b ^ p := by refine Nat.pow_le_pow_of_le_right h₀.2 ?_ exact le_of_lt hbp lemma imo_2022_p5_15 (a b p : ℕ) (h₀ : 0 < a ∧ 0 < b) (hp : Nat.Prime p) (h₁ : a ^ p = b ! + p) (hbp : p ≤ b) : (a, b, p) = (2, 2, 2) ∨ (a, b, p) = (3, 4, 3) := by have h₂: p ∣ a := by exact imo_2022_p5_3 a b p hp h₁ hbp by_cases hb2p: b < 2*p . have h₃: a = p := by exact imo_2022_p5_8 a b p h₀ hp h₁ hbp h₂ hb2p rw [h₃] at h₁ by_cases hp5: p < 5 . have h₄: 2 ≤ p := by exact Prime.two_le hp interval_cases p . left norm_num at h₁ have h₄: b.factorial = 2 := by linarith have g₅: (2:ℕ).factorial = 2 := by norm_num rw [← g₅] at h₄ have h₅: b = 2 := by refine (Nat.factorial_inj ?_).mp h₄ linarith rw [h₃,h₅] . right norm_num at h₁ rw [h₃] have h₄: b.factorial = 24 := by linarith have g₅: (4:ℕ).factorial = 24 := by exact rfl rw [← g₅] at h₄ have h₅: b = 4 := by refine (Nat.factorial_inj ?_).mp h₄ linarith rw [h₅] . exfalso contradiction . push_neg at hp5 exfalso -- lifting the exponent exact imo_2022_p5_12 b p hp hbp h₁ hp5 . push_neg at hb2p exfalso have h₃: p^2 ∣ a^p - b.factorial := by exact imo_2022_p5_13 a b p hp h₂ hb2p have g₃: b.factorial ≤ a^p := by exact le.intro (h₁.symm) have g₄: a^p - b.factorial = p := by rw [add_comm] at h₁ exact (Nat.sub_eq_iff_eq_add g₃).mpr h₁ have h₄: p^2 ∣ p := by rw [g₄] at h₃ exact h₃ have gp: 0 < p := by exact Prime.pos hp have h₅: p^2 ≤ p := by exact Nat.le_of_dvd gp h₄ have g₆: 1 < p := by exact Prime.one_lt hp have h₆: p^1 < p^2 := by exact Nat.pow_lt_pow_succ g₆ linarith lemma imo_2022_p5_15_1 (a b p : ℕ) (h₀ : 0 < a ∧ 0 < b) (hp : Nat.Prime p) (h₁ : a ^ p = b ! + p) (hbp : p ≤ b) (h₂ : p ∣ a) : (a, b, p) = (2, 2, 2) ∨ (a, b, p) = (3, 4, 3) := by by_cases hb2p: b < 2*p . have h₃: a = p := by exact imo_2022_p5_8 a b p h₀ hp h₁ hbp h₂ hb2p rw [h₃] at h₁ by_cases hp5: p < 5 . have h₄: 2 ≤ p := by exact Prime.two_le hp interval_cases p . left norm_num at h₁ have h₄: b.factorial = 2 := by linarith have g₅: (2:ℕ).factorial = 2 := by norm_num rw [← g₅] at h₄ have h₅: b = 2 := by refine (Nat.factorial_inj ?_).mp h₄ linarith rw [h₃,h₅] . right norm_num at h₁ rw [h₃] have h₄: b.factorial = 24 := by linarith have g₅: (4:ℕ).factorial = 24 := by exact rfl rw [← g₅] at h₄ have h₅: b = 4 := by refine (Nat.factorial_inj ?_).mp h₄ linarith rw [h₅] . exfalso contradiction . push_neg at hp5 exfalso -- lifting the exponent exact imo_2022_p5_12 b p hp hbp h₁ hp5 . push_neg at hb2p exfalso have h₃: p^2 ∣ a^p - b.factorial := by exact imo_2022_p5_13 a b p hp h₂ hb2p have g₃: b.factorial ≤ a^p := by exact le.intro (h₁.symm) have g₄: a^p - b.factorial = p := by rw [add_comm] at h₁ exact (Nat.sub_eq_iff_eq_add g₃).mpr h₁ have h₄: p^2 ∣ p := by rw [g₄] at h₃ exact h₃ have gp: 0 < p := by exact Prime.pos hp have h₅: p^2 ≤ p := by exact Nat.le_of_dvd gp h₄ have g₆: 1 < p := by exact Prime.one_lt hp have h₆: p^1 < p^2 := by exact Nat.pow_lt_pow_succ g₆ linarith lemma imo_2022_p5_15_2 (a b p : ℕ) (h₀ : 0 < a ∧ 0 < b) (hp : Nat.Prime p) (h₁ : a ^ p = b ! + p) (hbp : p ≤ b) (h₂ : p ∣ a) (hb2p : b < 2 * p) : (a, b, p) = (2, 2, 2) ∨ (a, b, p) = (3, 4, 3) := by have h₃: a = p := by exact imo_2022_p5_8 a b p h₀ hp h₁ hbp h₂ hb2p rw [h₃] at h₁ by_cases hp5: p < 5 . have h₄: 2 ≤ p := by exact Prime.two_le hp interval_cases p . left norm_num at h₁ have h₄: b.factorial = 2 := by linarith have g₅: (2:ℕ).factorial = 2 := by norm_num rw [← g₅] at h₄ have h₅: b = 2 := by refine (Nat.factorial_inj ?_).mp h₄ linarith rw [h₃,h₅] . right norm_num at h₁ rw [h₃] have h₄: b.factorial = 24 := by linarith have g₅: (4:ℕ).factorial = 24 := by exact rfl rw [← g₅] at h₄ have h₅: b = 4 := by refine (Nat.factorial_inj ?_).mp h₄ linarith rw [h₅] . exfalso contradiction . push_neg at hp5 exfalso -- lifting the exponent exact imo_2022_p5_12 b p hp hbp h₁ hp5 lemma imo_2022_p5_15_3 (a b p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) (hp : Nat.Prime p) (h₁ : a ^ p = b ! + p) (hbp : p ≤ b) (h₂ : p ∣ a) (hb2p : b < 2 * p) (h₃ : a = p) : (a, b, p) = (2, 2, 2) ∨ (a, b, p) = (3, 4, 3) := by rw [h₃] at h₁ by_cases hp5: p < 5 . have h₄: 2 ≤ p := by exact Prime.two_le hp interval_cases p . left norm_num at h₁ have h₄: b.factorial = 2 := by linarith have g₅: (2:ℕ).factorial = 2 := by norm_num rw [← g₅] at h₄ have h₅: b = 2 := by refine (Nat.factorial_inj ?_).mp h₄ linarith rw [h₃,h₅] . right norm_num at h₁ rw [h₃] have h₄: b.factorial = 24 := by linarith have g₅: (4:ℕ).factorial = 24 := by exact rfl rw [← g₅] at h₄ have h₅: b = 4 := by refine (Nat.factorial_inj ?_).mp h₄ linarith rw [h₅] . exfalso contradiction . push_neg at hp5 exfalso -- lifting the exponent exact imo_2022_p5_12 b p hp hbp h₁ hp5 lemma imo_2022_p5_15_4 (a b p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) (hp : Nat.Prime p) (h₁ : p ^ p = b ! + p) (hbp : p ≤ b) (h₂ : p ∣ a) (hb2p : b < 2 * p) (h₃ : a = p) (hp5 : p < 5) : (a, b, p) = (2, 2, 2) ∨ (a, b, p) = (3, 4, 3) := by have h₄: 2 ≤ p := by exact Prime.two_le hp interval_cases p . left norm_num at h₁ have h₄: b.factorial = 2 := by linarith have g₅: (2:ℕ).factorial = 2 := by norm_num rw [← g₅] at h₄ have h₅: b = 2 := by refine (Nat.factorial_inj ?_).mp h₄ linarith rw [h₃,h₅] . right norm_num at h₁ rw [h₃] have h₄: b.factorial = 24 := by linarith have g₅: (4:ℕ).factorial = 24 := by exact rfl rw [← g₅] at h₄ have h₅: b = 4 := by refine (Nat.factorial_inj ?_).mp h₄ linarith rw [h₅] . exfalso contradiction lemma imo_2022_p5_15_5 (a b p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) (hp : Nat.Prime p) (h₁ : p ^ p = b ! + p) (hbp : p ≤ b) (h₂ : p ∣ a) (hb2p : b < 2 * p) (h₃ : a = p) (hp5 : p < 5) (h₄ : 2 ≤ p) : (a, b, p) = (2, 2, 2) ∨ (a, b, p) = (3, 4, 3) := by interval_cases p . left norm_num at h₁ have h₄: b.factorial = 2 := by linarith have g₅: (2:ℕ).factorial = 2 := by norm_num rw [← g₅] at h₄ have h₅: b = 2 := by refine (Nat.factorial_inj ?_).mp h₄ linarith rw [h₃,h₅] . right norm_num at h₁ rw [h₃] have h₄: b.factorial = 24 := by linarith have g₅: (4:ℕ).factorial = 24 := by exact rfl rw [← g₅] at h₄ have h₅: b = 4 := by refine (Nat.factorial_inj ?_).mp h₄ linarith rw [h₅] . exfalso contradiction lemma imo_2022_p5_15_6 (a b : ℕ) -- (p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) -- (hp : Nat.Prime 2) (h₁ : 2 ^ 2 = b ! + 2) (hbp : 2 ≤ b) -- (h₂ : 2 ∣ a) -- (hb2p : b < 2 * 2) (h₃ : a = 2) : -- (hp5 : 2 < 5) -- (h₄ : 2 ≤ 2) : (a, b, 2) = (2, 2, 2) ∨ (a, b, 2) = (3, 4, 3) := by left norm_num at h₁ have h₄: b.factorial = 2 := by linarith have g₅: (2:ℕ).factorial = 2 := by norm_num rw [← g₅] at h₄ have h₅: b = 2 := by refine (Nat.factorial_inj ?_).mp h₄ linarith rw [h₃,h₅] lemma imo_2022_p5_15_7 (a b : ℕ) -- (p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) -- (hp : Nat.Prime 2) (hbp : 2 ≤ b) -- (h₂ : 2 ∣ a) -- (hb2p : b < 2 * 2) (h₃ : a = 2) -- (hp5 : 2 < 5) -- (h₄ : 2 ≤ 2) (h₁ : 2 = b !) : (a, b, 2) = (2, 2, 2) := by have h₄: b.factorial = 2 := by linarith have g₅: (2:ℕ).factorial = 2 := by norm_num rw [← g₅] at h₄ have h₅: b = 2 := by refine (Nat.factorial_inj ?_).mp h₄ linarith rw [h₃,h₅] lemma imo_2022_p5_15_8 -- (a p : ℕ) (b : ℕ) -- (h₀ : 0 < a ∧ 0 < b) -- (hp : Nat.Prime 2) (hbp : 2 ≤ b) -- (h₂ : 2 ∣ a) -- (hb2p : b < 2 * 2) -- (h₃ : a = 2) -- (hp5 : 2 < 5) -- (h4 : 2 ≤ 2) -- (h₁ : 2 = b !) (h₄ : b ! = 2!) : -- (g₅ : 2! = 2) : b = 2 := by refine (Nat.factorial_inj ?_).mp h₄ linarith lemma imo_2022_p5_15_9 (a b : ℕ) -- (p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) -- (hp : Nat.Prime 3) (h₁ : 3 ^ 3 = b ! + 3) (hbp : 3 ≤ b) -- (h₂ : 3 ∣ a) -- (hb2p : b < 2 * 3) (h₃ : a = 3) : -- (hp5 : 3 < 5) -- (h₄ : 2 ≤ 3) : (a, b, 3) = (2, 2, 2) ∨ (a, b, 3) = (3, 4, 3) := by right norm_num at h₁ rw [h₃] have h₄: b.factorial = 24 := by linarith have g₅: (4:ℕ).factorial = 24 := by exact rfl rw [← g₅] at h₄ have h₅: b = 4 := by refine (Nat.factorial_inj ?_).mp h₄ linarith rw [h₅] lemma imo_2022_p5_15_10 (a b : ℕ) -- (p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) -- (hp : Nat.Prime 3) (hbp : 3 ≤ b) -- (h₂ : 3 ∣ a) -- (hb2p : b < 2 * 3) (h₃ : a = 3) -- (hp5 : 3 < 5) -- (h₄ : 2 ≤ 3) (h₁ : 24 = b !) : (a, b, 3) = (3, 4, 3) := by rw [h₃] have h₄: b.factorial = 24 := by linarith have g₅: (4:ℕ).factorial = 24 := by exact rfl rw [← g₅] at h₄ have h₅: b = 4 := by refine (Nat.factorial_inj ?_).mp h₄ linarith rw [h₅] lemma imo_2022_p5_15_11 (b : ℕ) -- (a p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) -- (hp : Nat.Prime 3) (hbp : 3 ≤ b) -- (h₂ : 3 ∣ a) -- (hb2p : b < 2 * 3) -- (h₃ : a = 3) -- (hp5 : 3 < 5) -- (h4 : 2 ≤ 3) -- (h₁ : 24 = b !) (h₄ : b ! = 4!) : -- (g₅ : 4! = 24) : b = 4 := by refine (Nat.factorial_inj ?_).mp h₄ linarith lemma imo_2022_p5_15_12 (a b : ℕ) -- (p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) (hp : Nat.Prime 4) : -- (h₁ : 4 ^ 4 = b ! + 4) -- (hbp : 4 ≤ b) -- (h₂ : 4 ∣ a) -- (hb2p : b < 2 * 4) -- (h₃ : a = 4) -- (hp5 : 4 < 5) -- (h₄ : 2 ≤ 4) : (a, b, 4) = (2, 2, 2) ∨ (a, b, 4) = (3, 4, 3) := by exfalso contradiction lemma imo_2022_p5_15_13 (a b p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) (hp : Nat.Prime p) (h₁ : p ^ p = b ! + p) (hbp : p ≤ b) -- (h₂ : p ∣ a) -- (hb2p : b < 2 * p) -- (h₃ : a = p) (hp5 : 5 ≤ p) : (a, b, p) = (2, 2, 2) ∨ (a, b, p) = (3, 4, 3) := by exfalso -- lifting the exponent exact imo_2022_p5_12 b p hp hbp h₁ hp5 lemma imo_2022_p5_15_14 (a b p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) (hp : Nat.Prime p) (h₁ : a ^ p = b ! + p) -- (hbp : p ≤ b) (h₂ : p ∣ a) (hb2p : 2 * p ≤ b) : (a, b, p) = (2, 2, 2) ∨ (a, b, p) = (3, 4, 3) := by exfalso have h₃: p^2 ∣ a^p - b.factorial := by exact imo_2022_p5_13 a b p hp h₂ hb2p have g₃: b.factorial ≤ a^p := by exact le.intro (h₁.symm) have g₄: a^p - b.factorial = p := by rw [add_comm] at h₁ exact (Nat.sub_eq_iff_eq_add g₃).mpr h₁ have h₄: p^2 ∣ p := by rw [g₄] at h₃ exact h₃ have gp: 0 < p := by exact Prime.pos hp have h₅: p^2 ≤ p := by exact Nat.le_of_dvd gp h₄ have g₆: 1 < p := by exact Prime.one_lt hp have h₆: p^1 < p^2 := by exact Nat.pow_lt_pow_succ g₆ linarith lemma imo_2022_p5_15_15 (a b p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) (hp : Nat.Prime p) (h₁ : a ^ p = b ! + p) -- (hbp : p ≤ b) (h₂ : p ∣ a) (hb2p : 2 * p ≤ b) : False := by have h₃: p^2 ∣ a^p - b.factorial := by exact imo_2022_p5_13 a b p hp h₂ hb2p have g₃: b.factorial ≤ a^p := by exact le.intro (h₁.symm) have g₄: a^p - b.factorial = p := by rw [add_comm] at h₁ exact (Nat.sub_eq_iff_eq_add g₃).mpr h₁ have h₄: p^2 ∣ p := by rw [g₄] at h₃ exact h₃ have gp: 0 < p := by exact Prime.pos hp have h₅: p^2 ≤ p := by exact Nat.le_of_dvd gp h₄ have g₆: 1 < p := by exact Prime.one_lt hp have h₆: p^1 < p^2 := by exact Nat.pow_lt_pow_succ g₆ linarith lemma imo_2022_p5_15_16 (a b p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) (hp : Nat.Prime p) (h₁ : a ^ p = b ! + p) -- (hbp : p ≤ b) -- (h₂ : p ∣ a) -- (hb2p : 2 * p ≤ b) (h₃ : p ^ 2 ∣ a ^ p - b !) : False := by have g₃: b.factorial ≤ a^p := by exact le.intro (h₁.symm) have g₄: a^p - b.factorial = p := by rw [add_comm] at h₁ exact (Nat.sub_eq_iff_eq_add g₃).mpr h₁ have h₄: p^2 ∣ p := by rw [g₄] at h₃ exact h₃ have gp: 0 < p := by exact Prime.pos hp have h₅: p^2 ≤ p := by exact Nat.le_of_dvd gp h₄ have g₆: 1 < p := by exact Prime.one_lt hp have h₆: p^1 < p^2 := by exact Nat.pow_lt_pow_succ g₆ linarith lemma imo_2022_p5_15_17 (a b p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) (hp : Nat.Prime p) (h₁ : a ^ p = b ! + p) -- (hbp : p ≤ b) -- (h₂ : p ∣ a) -- (hb2p : 2 * p ≤ b) (h₃ : p ^ 2 ∣ a ^ p - b !) (g₃ : b ! ≤ a ^ p) : False := by have g₄: a^p - b.factorial = p := by rw [add_comm] at h₁ exact (Nat.sub_eq_iff_eq_add g₃).mpr h₁ have h₄: p^2 ∣ p := by rw [g₄] at h₃ exact h₃ have gp: 0 < p := by exact Prime.pos hp have h₅: p^2 ≤ p := by exact Nat.le_of_dvd gp h₄ have g₆: 1 < p := by exact Prime.one_lt hp have h₆: p^1 < p^2 := by exact Nat.pow_lt_pow_succ g₆ linarith lemma imo_2022_p5_15_18 (a b p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) -- (hp : Nat.Prime p) (h₁ : a ^ p = b ! + p) (hbp : p ≤ b) (h₂ : p ∣ a) (hb2p : 2 * p ≤ b) (h₃ : p ^ 2 ∣ a ^ p - b !) (g₃ : b ! ≤ a ^ p) : a ^ p - b ! = p := by rw [add_comm] at h₁ exact (Nat.sub_eq_iff_eq_add g₃).mpr h₁ lemma imo_2022_p5_15_19 (a b p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) (hp : Nat.Prime p) -- (h₁ : a ^ p = b ! + p) -- (hbp : p ≤ b) -- (h₂ : p ∣ a) -- (hb2p : 2 * p ≤ b) (h₃ : p ^ 2 ∣ a ^ p - b !) -- (g₃ : b ! ≤ a ^ p) (g₄ : a ^ p - b ! = p) : False := by have h₄: p^2 ∣ p := by rw [g₄] at h₃ exact h₃ have gp: 0 < p := by exact Prime.pos hp have h₅: p^2 ≤ p := by exact Nat.le_of_dvd gp h₄ have g₆: 1 < p := by exact Prime.one_lt hp have h₆: p^1 < p^2 := by exact Nat.pow_lt_pow_succ g₆ linarith lemma imo_2022_p5_15_20 (a b p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) -- (hp : Nat.Prime p) -- (h₁ : a ^ p = b ! + p) -- (hbp : p ≤ b) -- (h₂ : p ∣ a) -- (hb2p : 2 * p ≤ b) (h₃ : p ^ 2 ∣ a ^ p - b !) (g₃ : b ! ≤ a ^ p) (g₄ : a ^ p - b ! = p) : p ^ 2 ∣ p := by rw [g₄] at h₃ exact h₃ lemma imo_2022_p5_15_21 -- (a b : ℕ) (p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) (hp : Nat.Prime p) -- (h₁ : a ^ p = b ! + p) -- (hbp : p ≤ b) -- (h₂ : p ∣ a) -- (hb2p : 2 * p ≤ b) -- (h₃ : p ^ 2 ∣ a ^ p - b !) -- (g₃ : b ! ≤ a ^ p) -- (g₄ : a ^ p - b ! = p) (h₄ : p ^ 2 ∣ p) : False := by have gp: 0 < p := by exact Prime.pos hp have h₅: p^2 ≤ p := by exact Nat.le_of_dvd gp h₄ have g₆: 1 < p := by exact Prime.one_lt hp have h₆: p^1 < p^2 := by exact Nat.pow_lt_pow_succ g₆ linarith lemma imo_2022_p5_15_22 (a b p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) (hp : Nat.Prime p) -- (h₁ : a ^ p = b ! + p) -- (hbp : p ≤ b) -- (h₂ : p ∣ a) -- (hb2p : 2 * p ≤ b) (h₃ : p ^ 2 ∣ a ^ p - b !) -- (g₃ : b ! ≤ a ^ p) (g₄ : a ^ p - b ! = p) : p ^ 2 ≤ p := by have gp: 0 < p := by exact Prime.pos hp have h₄: p^2 ∣ p := by rw [g₄] at h₃ exact h₃ exact Nat.le_of_dvd gp h₄ lemma imo_2022_p5_15_23 -- (a b : ℕ) (p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) (hp : Nat.Prime p) -- (h₁ : a ^ p = b ! + p) -- (hbp : p ≤ b) -- (h₂ : p ∣ a) -- (hb2p : 2 * p ≤ b) -- (h₃ : p ^ 2 ∣ a ^ p - b !) -- (g₃ : b ! ≤ a ^ p) -- (g₄ : a ^ p - b ! = p) -- (h₄ : p ^ 2 ∣ p) -- (gp : 0 < p) (h₅ : p ^ 2 ≤ p) : False := by have g₆: 1 < p := by exact Prime.one_lt hp have h₆: p^1 < p^2 := by exact Nat.pow_lt_pow_succ g₆ linarith lemma imo_2022_p5_15_24 -- (a b : ℕ) (p : ℕ) -- (h₀ : 0 < a ∧ 0 < b) (hp : Nat.Prime p) : -- (h₁ : a ^ p = b ! + p) -- (hbp : p ≤ b) -- (h₂ : p ∣ a) -- (hb2p : 2 * p ≤ b) -- (h₃ : p ^ 2 ∣ a ^ p - b !) -- (g₃ : b ! ≤ a ^ p) -- (g₄ : a ^ p - b ! = p) -- (h₄ : p ^ 2 ∣ p) -- (h₅ : p ^ 2 ≤ p) : p ^ 1 < p ^ 2 := by refine Nat.pow_lt_pow_succ ?_ exact Prime.one_lt hp