diff --git "a/Lemmas/imo_2022_p5_lemmas.lean" "b/Lemmas/imo_2022_p5_lemmas.lean" new file mode 100644--- /dev/null +++ "b/Lemmas/imo_2022_p5_lemmas.lean" @@ -0,0 +1,6777 @@ +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) + -- (h��� : (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) + (h��� : 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) + (h��� : 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