{"claim":{"bundle":{"code":"","commit":"e3af5bf76817c3c296c14a8462c1ba439e31bc4e","config":null,"environment":"lemma/0.6.1 agent","logs":"","seeds":[]},"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","Lemma.primes_mod_6_eq_5":"import Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity\n\nnamespace Lemma.P65\n\ntheorem exists_factor (N : ℕ) (hN : N % 6 = 5) :\n    ∃ q, q.Prime ∧ q % 6 = 5 ∧ q ∣ N := 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    set q := N.minFac with hqdef\n    have hq2 : q % 2 = 1 := by\n      by_contra h\n      have h2 : 2 ∣ q := by omega\n      have : q = 2 := ((Nat.prime_dvd_prime_iff_eq Nat.prime_two hq).1 h2).symm\n      rw [this] at hqd\n      omega\n    have hq3 : q % 3 ≠ 0 := by\n      intro h\n      have h3 : 3 ∣ q := by omega\n      have : q = 3 := ((Nat.prime_dvd_prime_iff_eq Nat.prime_three hq).1 h3).symm\n      rw [this] at hqd\n      omega\n    by_cases h5 : q % 6 = 5\n    · exact ⟨q, hq, h5, hqd⟩\n    · have h1 : q % 6 = 1 := by omega\n      obtain ⟨m, hm⟩ := hqd\n      have hmod := Nat.mul_mod q m 6\n      rw [h1, ← hm, hN] at hmod\n      have hm5 : m % 6 = 5 := by omega\n      have hq2' := hq.two_le\n      have hmlt : m < N := by\n        have hm0 : 0 < m := by\n          apply Nat.pos_of_ne_zero\n          rintro rfl\n          rw [mul_zero] at hm\n          omega\n        rw [hm]\n        calc m = 1 * m := (one_mul m).symm\n          _ < q * m := Nat.mul_lt_mul_of_pos_right hq.one_lt hm0\n      obtain ⟨r, hr, hr5, hrd⟩ := ih m hmlt hm5\n      exact ⟨r, hr, hr5, hrd.trans ⟨q, by rw [hm, mul_comm]⟩⟩\n\nend Lemma.P65\n\ntheorem Lemma.primes_mod_6_eq_5 :\n    Set.Infinite {p : ℕ | p.Prime ∧ p % 6 = 5} := by\n  apply Set.infinite_of_forall_exists_gt\n  intro n\n  set x := (n + 3).factorial with hxdef\n  have hx0 : 0 < x := Nat.factorial_pos _\n  obtain ⟨q, hq, h5, hd⟩ := Lemma.P65.exists_factor (6 * x - 1) (by omega)\n  refine ⟨q, ⟨hq, h5⟩, ?_⟩\n  by_contra hle\n  push_neg at hle\n  have hqx : q ∣ 6 * x := Dvd.dvd.mul_left (Nat.dvd_factorial hq.pos (by omega)) 6\n  have h1 : q ∣ 6 * x - (6 * x - 1) := Nat.dvd_sub hqx hd\n  have : 6 * x - (6 * x - 1) = 1 := by omega\n  rw [this] at h1\n  exact hq.one_lt.ne' (Nat.dvd_one.1 h1)\n","Lemma.primes_mod_7_eq_6":"import Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity\n\nnamespace Lemma.P76\n\nopen Polynomial\n\ntheorem prime7 : Nat.Prime 7 := by decide\n\ntheorem key (q : ℕ) (hq : q.Prime) (hq7 : q ≠ 7) (t : ZMod q)\n    (ht : t ^ 3 + t ^ 2 - 2 * t - 1 = 0) : q % 7 = 1 ∨ q % 7 = 6 := by\n  haveI := Fact.mk hq\n  haveI : Fact (Nat.Prime 7) := Fact.mk prime7\n  set p : (ZMod q)[X] := X ^ 2 - C t * X + 1 with hpdef\n  have hp : p.natDegree = 2 := by rw [hpdef]; compute_degree!\n  have hpu : ¬ IsUnit p := by\n    intro h\n    have := Polynomial.natDegree_eq_zero_of_isUnit h\n    omega\n  have hp0 : p ≠ 0 := by\n    intro h\n    rw [h] at hp\n    simp at hp\n  obtain ⟨g, hg, hgp⟩ := WfDvdMonoid.exists_irreducible_factor hpu hp0\n  haveI := Fact.mk hg\n  let K := AdjoinRoot g\n  let r : K := AdjoinRoot.root g\n  have hr0 : aeval r p = 0 := by\n    obtain ⟨c, hc⟩ := hgp\n    rw [hc, map_mul, AdjoinRoot.aeval_eq, AdjoinRoot.mk_self, zero_mul]\n  set T : K := algebraMap (ZMod q) K t with hTdef\n  have hr : r ^ 2 - T * r + 1 = 0 := by\n    rw [hpdef] at hr0\n    simpa [hTdef] using hr0\n  haveI : CharP K q := charP_of_injective_algebraMap (algebraMap (ZMod q) K).injective q\n  have hT : T ^ 3 + T ^ 2 - 2 * T - 1 = 0 := by\n    have := congrArg (algebraMap (ZMod q) K) ht\n    simpa [hTdef, map_ofNat] using this\n  have hTq : T ^ q = T := by rw [hTdef, ← map_pow, ZMod.pow_card]\n  have hr7 : r ^ 7 = 1 := by\n    linear_combination (r - 1) * (r ^ 3 * hT + (1 + r + r * T + r ^ 2 * T + r ^ 2 * T ^ 2\n      + r ^ 3 + r ^ 3 * T + r ^ 4) * hr)\n  have hr1 : r ≠ 1 := by\n    intro h1\n    rw [h1] at hr\n    have hT2 : T = 2 := by linear_combination -hr\n    rw [hT2] at hT\n    have h7 : ((7 : ℕ) : K) = 0 := by push_cast; linear_combination hT\n    have hq7' : q ∣ 7 := (CharP.cast_eq_zero_iff K q 7).1 h7\n    exact hq7 ((Nat.prime_dvd_prime_iff_eq hq prime7).1 hq7')\n  have hrne : r ≠ 0 := by\n    intro h0\n    rw [h0] at hr\n    simp at hr\n  have hrq : (r ^ q) ^ 2 - T * r ^ q + 1 = 0 := by\n    have : (r ^ 2 - T * r + 1) ^ q = 0 := by rw [hr, zero_pow hq.ne_zero]\n    rw [add_pow_char, sub_pow_char, mul_pow, hTq, one_pow] at this\n    rw [← this]\n    ring\n  have hprod : (r ^ q - r) * (r ^ q - (T - r)) = 0 := by linear_combination hrq - hr\n  have hord : orderOf r = 7 := orderOf_eq_prime hr7 hr1\n  have hq2 := hq.two_le\n  rcases mul_eq_zero.1 hprod with h | h\n  · have h' : r ^ (q - 1) = 1 := by\n      have e : r ^ q = r ^ (q - 1) * r := by\n        rw [← pow_succ]; congr 1; omega\n      have : (r ^ (q - 1) - 1) * r = 0 := by linear_combination e.symm.trans (sub_eq_zero.1 h)\n      rcases mul_eq_zero.1 this with h2 | h2\n      · exact sub_eq_zero.1 h2\n      · exact absurd h2 hrne\n    have := orderOf_dvd_of_pow_eq_one h'\n    rw [hord] at this\n    omega\n  · have h' : r ^ (q + 1) = 1 := by\n      rw [pow_succ, sub_eq_zero.1 h]\n      linear_combination -hr\n    have := orderOf_dvd_of_pow_eq_one h'\n    rw [hord] at this\n    omega\n\ntheorem exists_factor (N : ℕ) (hN : N % 7 = 6)\n    (hall : ∀ q, q.Prime → q ∣ N → q % 7 = 1 ∨ q % 7 = 6) :\n    ∃ q, q.Prime ∧ q % 7 = 6 ∧ q ∣ N := 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    set q := N.minFac with hqdef\n    rcases hall q hq hqd with h1 | h6\n    · obtain ⟨m, hm⟩ := hqd\n      have hmod := Nat.mul_mod q m 7\n      rw [h1, ← hm, hN] at hmod\n      have hm6 : m % 7 = 6 := by omega\n      have hmlt : m < N := by\n        have hm0 : 0 < m := by\n          apply Nat.pos_of_ne_zero\n          rintro rfl\n          rw [mul_zero] at hm\n          omega\n        rw [hm]\n        calc m = 1 * m := (one_mul m).symm\n          _ < q * m := Nat.mul_lt_mul_of_pos_right hq.one_lt hm0\n      have hall' : ∀ r, r.Prime → r ∣ m → r % 7 = 1 ∨ r % 7 = 6 := fun r hr hrm =>\n        hall r hr (hrm.trans ⟨q, by rw [hm, mul_comm]⟩)\n      obtain ⟨r, hr, hr6, hrd⟩ := ih m hmlt hm6 hall'\n      exact ⟨r, hr, hr6, hrd.trans ⟨q, by rw [hm, mul_comm]⟩⟩\n    · exact ⟨q, hq, h6, hqd⟩\n\nend Lemma.P76\n\ntheorem Lemma.primes_mod_7_eq_6 :\n    Set.Infinite {p : ℕ | p.Prime ∧ p % 7 = 6} := by\n  apply Set.infinite_of_forall_exists_gt\n  intro n\n  set y := (n + 7).factorial with hydef\n  have hy0 : 0 < y := Nat.factorial_pos _\n  set x := 7 * y with hxdef\n  have hle : 2 * x + 1 ≤ x ^ 3 + x ^ 2 := by\n    have : 7 ≤ x := by omega\n    nlinarith\n  set N := x ^ 3 + x ^ 2 - (2 * x + 1) with hNdef\n  have hN : N + (2 * x + 1) = x ^ 3 + x ^ 2 := Nat.sub_add_cancel hle\n  have hN' : N + (14 * y + 1) = 343 * y ^ 3 + 49 * y ^ 2 := by\n    rw [hxdef] at hN; linarith [hN]\n  have hN7 : N % 7 = 6 := by\n    generalize y ^ 3 = a at hN'\n    generalize y ^ 2 = b at hN'\n    omega\n  have hfac : ∀ q, q.Prime → q ∣ N → (q % 7 = 1 ∨ q % 7 = 6) ∧ ¬ q ∣ y := by\n    intro q hq hqN\n    haveI := Fact.mk hq\n    have hc : ((N : ℕ) : ZMod q) = 0 := (CharP.cast_eq_zero_iff (ZMod q) q _).2 hqN\n    have hc2 : ((N + (2 * x + 1) : ℕ) : ZMod q) = ((x ^ 3 + x ^ 2 : ℕ) : ZMod q) := by rw [hN]\n    push_cast at hc2\n    rw [hc] at hc2\n    have hf : (x : ZMod q) ^ 3 + (x : ZMod q) ^ 2 - 2 * (x : ZMod q) - 1 = 0 := by\n      linear_combination -hc2\n    have hny : ¬ q ∣ y := by\n      intro hqy\n      have : (x : ZMod q) = 0 := by\n        rw [hxdef]; push_cast\n        rw [(CharP.cast_eq_zero_iff (ZMod q) q _).2 hqy, mul_zero]\n      rw [this] at hf\n      norm_num at hf\n    refine ⟨Lemma.P76.key q hq ?_ _ hf, hny⟩\n    rintro rfl\n    omega\n  obtain ⟨q, hq, h6, hqd⟩ := Lemma.P76.exists_factor N hN7 (fun q hq hd => (hfac q hq hd).1)\n  refine ⟨q, ⟨hq, h6⟩, ?_⟩\n  by_contra hle'\n  push_neg at hle'\n  exact (hfac q hq hqd).2 (Nat.dvd_factorial hq.pos (by omega))\n"}},"flags":[],"id":15,"label":"network-best","measured":4.0,"reason":null,"score":4.0,"significance":[],"status":"verified","verdicts":[]},"entry":{"body":"Prove Lemma.primes_mod_7_eq_6 under its imports\nMethod: Write a complete Lean proof of Lemma.primes_mod_7_eq_6: Set.Infinite {p : ℕ | p.Prime ∧ p % 7 = 6}. Start from proofs/Lemma.primes_mod_7_eq_6.lean. If it is out of reach in the budget, prove a useful lemma toward it, post what blocks you to the chat, and submit what you have verified.\nThreshold: 3 (must beat). Result 4; checked by the coordinator (tier 1, exact).\nSubmitter notes (data): primes_mod_7_eq_6 via the cubic f(t)=t^3+t^2-2t-1 (min poly of 2cos(2π/7)). x=7(n+7)!, N=f(x)≡6 mod 7. For prime q|N, q≠7: take an irreducible factor g of X^2-tX+1 over ZMod q, K=AdjoinRoot g (field), r=root; r^7=1 (Φ7(r)=r^3 f(t)+(r^2-tr+1)Q), r≠1, so orderOf r=7; Frobenius gives r^q root of same quadratic so r^q∈{r, t-r}, hence 7 | q-1 or q+1. Strong induction gives a factor ≡6; q>n since q∤(n+7)!.","claim":15,"created":1791838017,"credit":[2],"id":15,"kind":"result","label":"network-best","parent":11,"program":"euclid-primes","score":4.0,"title":"Network best: theorems proved 4","unit":43},"followups":[{"hypothesis":"Prove Lemma.primes_mod_8_eq_1 under its imports","status":"assigned","unit":44}]}