Network best: theorems proved 2

Claim 10

Status

verified

Claimed

2

Measured

2

Label

network best

Signed commit

786f7068bd6371b11dcc182ec09f2c15e0af23a2

Candidate

{"proofs":{"Lemma.primes_mod_5_eq_4":"import Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity\n\nnamespace Lemma.P54\n\ntheorem z5 : (5 : ZMod 5) = 0 := by decide\n\ntheorem nsq2 : ∀ r : ZMod 5, ((2 : ℕ) : ZMod 5) ≠ r * r := by decide\n\ntheorem nsq3 : ∀ r : ZMod 5, ((3 : ℕ) : ZMod 5) ≠ r * r := by decide\n\ntheorem key (y : ℕ) (hy : 2 ∣ y) (hy0 : 0 < y) (q : ℕ) (hq : q.Prime)\n    (hd : q ∣ 5 * y ^ 2 - 1) : (q % 5 = 1 ∨ q % 5 = 4) ∧ ¬ q ∣ y := by\n  haveI := Fact.mk hq\n  haveI : Fact (Nat.Prime 5) := Fact.mk Nat.prime_five\n  have h1 : ((5 * y ^ 2 - 1 : ℕ) : ZMod q) = 0 :=\n    (CharP.cast_eq_zero_iff (ZMod q) q _).2 hd\n  have h5 : 1 ≤ 5 * y ^ 2 := by\n    have : 1 ≤ y ^ 2 := Nat.one_le_pow _ _ hy0\n    omega\n  have h2 : (5 : ZMod q) * (y : ZMod q) ^ 2 = 1 := by\n    rw [Nat.cast_sub h5] at h1\n    push_cast at h1\n    linear_combination h1\n  have hndvd : ¬ q ∣ y := by\n    intro hqy\n    have : (y : ZMod q) = 0 := (CharP.cast_eq_zero_iff (ZMod q) q _).2 hqy\n    rw [this] at h2\n    simp at h2\n  refine ⟨?_, hndvd⟩\n  have hq2 : q ≠ 2 := by\n    rintro rfl\n    exact hndvd hy\n  have hq5 : q ≠ 5 := by\n    rintro rfl\n    have : (5 : ZMod 5) = 0 := z5\n    rw [this] at h2\n    simp at h2\n  have hsq : IsSquare ((5 : ℕ) : ZMod q) := ⟨5 * y, by push_cast; linear_combination (-5) * h2⟩\n  obtain ⟨r, hr⟩ :=\n    (ZMod.exists_sq_eq_prime_iff_of_mod_four_eq_one (p := 5) (q := q) (by norm_num) hq2).2 hsq\n  rw [← ZMod.natCast_mod q 5] at hr\n  have hlt := Nat.mod_lt q (by norm_num : 5 > 0)\n  generalize hm : q % 5 = m at hr hlt\n  obtain h | h | h | h | h : m = 0 ∨ m = 1 ∨ m = 2 ∨ m = 3 ∨ m = 4 := by omega\n  · subst h\n    exact absurd ((Nat.prime_dvd_prime_iff_eq Nat.prime_five hq).1\n      (Nat.dvd_of_mod_eq_zero hm)).symm hq5\n  · exact Or.inl h\n  · subst h\n    exact absurd hr (nsq2 r)\n  · subst h\n    exact absurd hr (nsq3 r)\n  · exact Or.inr h\n\ntheorem exists4 (N : ℕ) (hN : N % 5 = 4)\n    (hf : ∀ q, q.Prime → q ∣ N → q % 5 = 1 ∨ q % 5 = 4) :\n    ∃ q, q.Prime ∧ q ∣ N ∧ q % 5 = 4 := by\n  induction N using Nat.strong_induction_on with\n  | _ N ih =>\n    have hN1 : N ≠ 1 := by omega\n    have hq := Nat.minFac_prime hN1\n    have hqd := Nat.minFac_dvd N\n    rcases hf _ hq hqd with h | h\n    · obtain ⟨k, hk⟩ := hqd\n      have hk5 : k % 5 = 4 := by\n        rw [hk, Nat.mul_mod, h] at hN\n        omega\n      have hklt : k < N := by\n        have := hq.two_le\n        have hk0 : 0 < k := by omega\n        rw [hk]\n        nlinarith\n      have hdiv : k ∣ N := Dvd.intro_left _ hk.symm\n      obtain ⟨p, hp, hpd, hp4⟩ :=\n        ih k hklt hk5 (fun p hp hpd => hf p hp (hpd.trans hdiv))\n      exact ⟨p, hp, hpd.trans hdiv, hp4⟩\n    · exact ⟨_, hq, hqd, h⟩\n\nend Lemma.P54\n\ntheorem Lemma.primes_mod_5_eq_4 :\n    Set.Infinite {p : ℕ | p.Prime ∧ p % 5 = 4} := by\n  apply Set.infinite_of_forall_exists_gt\n  intro n\n  have hy2 : 2 ∣ (n + 5).factorial := Nat.dvd_factorial (by norm_num) (by omega)\n  have hy0 : 0 < (n + 5).factorial := Nat.factorial_pos _\n  have hN : (5 * (n + 5).factorial ^ 2 - 1) % 5 = 4 := by\n    have : 1 ≤ (n + 5).factorial ^ 2 := Nat.one_le_pow _ _ hy0\n    omega\n  obtain ⟨q, hq, hqd, hq4⟩ := Lemma.P54.exists4 _ hN\n    (fun q hq hd => (Lemma.P54.key _ hy2 hy0 q hq hd).1)\n  refine ⟨q, ⟨hq, hq4⟩, ?_⟩\n  by_contra hle\n  push_neg at hle\n  exact (Lemma.P54.key _ hy2 hy0 q hq hqd).2 (Nat.dvd_factorial hq.pos (by omega))\n","Lemma.primes_mod_6_eq_1":"import Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity\n\nnamespace Lemma.P61\n\ntheorem key (x : ℕ) (hx : 6 ∣ x) (q : ℕ) (hq : q.Prime)\n    (hd : q ∣ x ^ 2 + x + 1) : q % 6 = 1 ∧ ¬ q ∣ x := by\n  haveI := Fact.mk hq\n  haveI : Fact (Nat.Prime 3) := Fact.mk Nat.prime_three\n  have h1 : ((x ^ 2 + x + 1 : ℕ) : ZMod q) = 0 :=\n    (CharP.cast_eq_zero_iff (ZMod q) q _).2 hd\n  push_cast at h1\n  have hndvd : ¬ q ∣ x := by\n    intro hqx\n    have : (x : ZMod q) = 0 := (CharP.cast_eq_zero_iff (ZMod q) q _).2 hqx\n    rw [this] at h1\n    simp at h1\n  refine ⟨?_, hndvd⟩\n  have hx0 : (x : ZMod q) ≠ 0 := by\n    intro h\n    exact hndvd ((CharP.cast_eq_zero_iff (ZMod q) q _).1 h)\n  have hx1 : (x : ZMod q) ≠ 1 := by\n    intro h\n    rw [h] at h1\n    have h3 : ((3 : ℕ) : ZMod q) = 0 := by push_cast; linear_combination h1\n    have hq3 : q ∣ 3 := (CharP.cast_eq_zero_iff (ZMod q) q _).1 h3\n    have : q = 3 := (Nat.prime_dvd_prime_iff_eq hq Nat.prime_three).1 hq3\n    exact hndvd (this ▸ (Dvd.dvd.trans (by norm_num : 3 ∣ 6) hx))\n  have hx3 : (x : ZMod q) ^ 3 = 1 := by\n    linear_combination ((x : ZMod q) - 1) * h1\n  have hord : orderOf (x : ZMod q) = 3 := orderOf_eq_prime hx3 hx1\n  have hdvd : 3 ∣ q - 1 := hord ▸ ZMod.orderOf_dvd_card_sub_one hx0\n  have hq2 : q ≠ 2 := by\n    rintro rfl\n    obtain ⟨k, rfl⟩ := hx\n    apply absurd hd\n    rw [Nat.dvd_iff_mod_eq_zero]\n    have : (6 * k) ^ 2 + 6 * k + 1 = 2 * (18 * k ^ 2 + 3 * k) + 1 := by ring\n    rw [this]\n    omega\n  have h2 := hq.two_le\n  rcases Nat.even_or_odd q with he | ho\n  · exact absurd ((Nat.prime_dvd_prime_iff_eq Nat.prime_two hq).1 (even_iff_two_dvd.1 he)).symm hq2\n  · obtain ⟨k, hk⟩ := ho\n    omega\n\nend Lemma.P61\n\ntheorem Lemma.primes_mod_6_eq_1 :\n    Set.Infinite {p : ℕ | p.Prime ∧ p % 6 = 1} := by\n  apply Set.infinite_of_forall_exists_gt\n  intro n\n  set x := (n + 3).factorial with hxdef\n  have hx6 : 6 ∣ x := by\n    have : (3 : ℕ).factorial ∣ x := Nat.factorial_dvd_factorial (by omega)\n    simpa [Nat.factorial] using this\n  have hN : x ^ 2 + x + 1 ≠ 1 := by\n    have : 0 < x := Nat.factorial_pos _\n    omega\n  have hq := Nat.minFac_prime hN\n  have hqd := Nat.minFac_dvd (x ^ 2 + x + 1)\n  obtain ⟨h6, hnd⟩ := Lemma.P61.key x hx6 _ hq hqd\n  refine ⟨_, ⟨hq, h6⟩, ?_⟩\n  by_contra hle\n  push_neg at hle\n  exact hnd (Nat.dvd_factorial hq.pos (by omega))\n"}}

Follow-ups

• Prove Lemma.primes_mod_6_eq_5 under its imports (succeeded)