internlm/Lean-Github huggingface dataset:
Theorem: gcd1or3
Proof state before:
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
⊢ g = 3 ^ k
Tactic:
apply Nat.eq_prime_pow_of_unique_prime_dvd
Proof state after:
case hpos
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
⊢ g ≠ 0

case h
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
⊢ ∀ {d : ℕ}, d.Prime → d ∣ g → d = 3
Theorem: gcd1or3
Proof state before:
case h
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
⊢ ∀ {d : ℕ}, d.Prime → d ∣ g → d = 3
Tactic:
intro d hdprime hddvdg
Proof state after:
case h
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
d : ℕ
hdprime : d.Prime
hddvdg : d ∣ g
⊢ d = 3
Theorem: gcd1or3
Proof state before:
case h
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
d : ℕ
hdprime : d.Prime
hddvdg : d ∣ g
⊢ d = 3
Tactic:
rw [← Int.natCast_dvd_natCast] at hddvdg
Proof state after:
case h
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
d : ℕ
hdprime : d.Prime
hddvdg : ↑d ∣ ↑g
⊢ d = 3
Theorem: gcd1or3
Proof state before:
case h
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
d : ℕ
hdprime : d.Prime
hddvdg : ↑d ∣ ↑g
⊢ d = 3
Tactic:
apply basic _ hdprime <;> apply dvd_trans hddvdg <;> rw [hg']
Proof state after:
case h.a
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
d : ℕ
hdprime : d.Prime
hddvdg : ↑d ∣ ↑g
⊢ ↑((2 * p).gcd (p ^ 2 + 3 * q ^ 2)) ∣ 2 * p

case h.a
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
d : ℕ
hdprime : d.Prime
hddvdg : ↑d ∣ ↑g
⊢ ↑((2 * p).gcd (p ^ 2 + 3 * q ^ 2)) ∣ p ^ 2 + 3 * q ^ 2
Theorem: gcd1or3
Proof state before:
case h.a
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
d : ℕ
hdprime : d.Prime
hddvdg : ↑d ∣ ↑g
⊢ ↑((2 * p).gcd (p ^ 2 + 3 * q ^ 2)) ∣ 2 * p

case h.a
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
d : ℕ
hdprime : d.Prime
hddvdg : ↑d ∣ ↑g
⊢ ↑((2 * p).gcd (p ^ 2 + 3 * q ^ 2)) ∣ p ^ 2 + 3 * q ^ 2
Tactic:
exacts[Int.gcd_dvd_left, Int.gcd_dvd_right]
Proof state after:
no goals
Theorem: gcd1or3
Proof state before:
case hpos
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
⊢ g ≠ 0
Tactic:
apply ne_of_gt
Proof state after:
case hpos.h
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
⊢ 0 < g
Theorem: gcd1or3
Proof state before:
case hpos.h
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
⊢ 0 < g
Tactic:
apply Nat.gcd_pos_of_pos_left
Proof state after:
case hpos.h.mpos
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
⊢ 0 < (2 * p).natAbs
Theorem: gcd1or3
Proof state before:
case hpos.h.mpos
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
⊢ 0 < (2 * p).natAbs
Tactic:
rw [Int.natAbs_mul]
Proof state after:
case hpos.h.mpos
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
⊢ 0 < Int.natAbs 2 * p.natAbs
Theorem: gcd1or3
Proof state before:
case hpos.h.mpos
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
⊢ 0 < Int.natAbs 2 * p.natAbs
Tactic:
apply mul_pos zero_lt_two
Proof state after:
case hpos.h.mpos
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
⊢ 0 < p.natAbs
Theorem: gcd1or3
Proof state before:
case hpos.h.mpos
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
⊢ 0 < p.natAbs
Tactic:
rwa [pos_iff_ne_zero, Int.natAbs_ne_zero]
Proof state after:
no goals
Theorem: gcd1or3
Proof state before:
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
hg : g = 3 ^ 2 * 3 ^ (k - 2)
H : 2 ≤ k
⊢ ¬IsUnit 3
Tactic:
rw [Int.isUnit_iff_natAbs_eq]
Proof state after:
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
hg : g = 3 ^ 2 * 3 ^ (k - 2)
H : 2 ≤ k
⊢ ¬Int.natAbs 3 = 1
Theorem: gcd1or3
Proof state before:
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
hg : g = 3 ^ 2 * 3 ^ (k - 2)
H : 2 ≤ k
⊢ ¬Int.natAbs 3 = 1
Tactic:
norm_num
Proof state after:
no goals
Theorem: gcd1or3
Proof state before:
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
hg : g = 3 ^ 2 * 3 ^ (k - 2)
H : 2 ≤ k
this : ¬IsUnit 3
⊢ 3 ∣ p
Tactic:
suffices 3 ∣ 2 * p
  by
  apply Int.dvd_mul_cancel_prime' _ dvd_rfl Int.prime_two this
  norm_num
Proof state after:
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
hg : g = 3 ^ 2 * 3 ^ (k - 2)
H : 2 ≤ k
this : ¬IsUnit 3
⊢ 3 ∣ 2 * p
Theorem: gcd1or3
Proof state before:
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
hg : g = 3 ^ 2 * 3 ^ (k - 2)
H : 2 ≤ k
this : ¬IsUnit 3
⊢ 3 ∣ 2 * p
Tactic:
have : 3 ∣ (g : ℤ) := by
  rw [hg, pow_two, mul_assoc, Int.ofNat_mul]
  apply dvd_mul_right
Proof state after:
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
hg : g = 3 ^ 2 * 3 ^ (k - 2)
H : 2 ≤ k
this✝ : ¬IsUnit 3
this : 3 ∣ ↑g
⊢ 3 ∣ 2 * p
Theorem: gcd1or3
Proof state before:
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
hg : g = 3 ^ 2 * 3 ^ (k - 2)
H : 2 ≤ k
this✝ : ¬IsUnit 3
this : 3 ∣ ↑g
⊢ 3 ∣ 2 * p
Tactic:
exact dvd_trans this Int.gcd_dvd_left
Proof state after:
no goals
Theorem: gcd1or3
Proof state before:
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
hg : g = 3 ^ 2 * 3 ^ (k - 2)
H : 2 ≤ k
this✝ : ¬IsUnit 3
this : 3 ∣ 2 * p
⊢ 3 ∣ p
Tactic:
apply Int.dvd_mul_cancel_prime' _ dvd_rfl Int.prime_two this
Proof state after:
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
hg : g = 3 ^ 2 * 3 ^ (k - 2)
H : 2 ≤ k
this✝ : ¬IsUnit 3
this : 3 ∣ 2 * p
⊢ ¬2 ∣ 3
Theorem: gcd1or3
Proof state before:
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
hg : g = 3 ^ 2 * 3 ^ (k - 2)
H : 2 ≤ k
this✝ : ¬IsUnit 3
this : 3 ∣ 2 * p
⊢ ¬2 ∣ 3
Tactic:
norm_num
Proof state after:
no goals
Theorem: gcd1or3
Proof state before:
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
hg : g = 3 ^ 2 * 3 ^ (k - 2)
H : 2 ≤ k
this : ¬IsUnit 3
⊢ 3 ∣ ↑g
Tactic:
rw [hg, pow_two, mul_assoc, Int.ofNat_mul]
Proof state after:
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
hg : g = 3 ^ 2 * 3 ^ (k - 2)
H : 2 ≤ k
this : ¬IsUnit 3
⊢ 3 ∣ ↑3 * ↑(3 * 3 ^ (k - 2))
Theorem: gcd1or3
Proof state before:
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
hg : g = 3 ^ 2 * 3 ^ (k - 2)
H : 2 ≤ k
this : ¬IsUnit 3
⊢ 3 ∣ ↑3 * ↑(3 * 3 ^ (k - 2))
Tactic:
apply dvd_mul_right
Proof state after:
no goals
Theorem: gcd1or3
Proof state before:
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
hg : g = 3 ^ 2 * 3 ^ (k - 2)
H : 2 ≤ k
this : ¬IsUnit 3
hdvdp : 3 ∣ p
⊢ 3 ∣ q
Tactic:
rw [← Int.pow_dvd_pow_iff two_ne_zero] at hdvdp
Proof state after:
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
hg : g = 3 ^ 2 * 3 ^ (k - 2)
H : 2 ≤ k
this : ¬IsUnit 3
hdvdp : 3 ^ 2 ∣ p ^ 2
⊢ 3 ∣ q
Theorem: gcd1or3
Proof state before:
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
hg : g = 3 ^ 2 * 3 ^ (k - 2)
H : 2 ≤ k
this : ¬IsUnit 3
hdvdp : 3 ^ 2 ∣ p ^ 2
⊢ 3 ∣ q
Tactic:
apply Prime.dvd_of_dvd_pow Int.prime_three
Proof state after:
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
hg : g = 3 ^ 2 * 3 ^ (k - 2)
H : 2 ≤ k
this : ¬IsUnit 3
hdvdp : 3 ^ 2 ∣ p ^ 2
⊢ 3 ∣ q ^ ?m.72510

p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
hg : g = 3 ^ 2 * 3 ^ (k - 2)
H : 2 ≤ k
this : ¬IsUnit 3
hdvdp : 3 ^ 2 ∣ p ^ 2
⊢ ℕ
Theorem: gcd1or3
Proof state before:
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
hg : g = 3 ^ 2 * 3 ^ (k - 2)
H : 2 ≤ k
this : ¬IsUnit 3
hdvdp : 3 ^ 2 ∣ p ^ 2
⊢ 3 ∣ q ^ ?m.72510

p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
hg : g = 3 ^ 2 * 3 ^ (k - 2)
H : 2 ≤ k
this : ¬IsUnit 3
hdvdp : 3 ^ 2 ∣ p ^ 2
⊢ ℕ
Tactic:
rw [← mul_dvd_mul_iff_left (three_ne_zero' ℤ), ← pow_two, ← dvd_add_right hdvdp]
Proof state after:
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
hg : g = 3 ^ 2 * 3 ^ (k - 2)
H : 2 ≤ k
this : ¬IsUnit 3
hdvdp : 3 ^ 2 ∣ p ^ 2
⊢ 3 ^ 2 ∣ p ^ 2 + 3 * q ^ ?m.72510

p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
hg : g = 3 ^ 2 * 3 ^ (k - 2)
H : 2 ≤ k
this : ¬IsUnit 3
hdvdp : 3 ^ 2 ∣ p ^ 2
⊢ ℕ

p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
hg : g = 3 ^ 2 * 3 ^ (k - 2)
H : 2 ≤ k
this : ¬IsUnit 3
hdvdp : 3 ^ 2 ∣ p ^ 2
⊢ ℕ

p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
hg : g = 3 ^ 2 * 3 ^ (k - 2)
H : 2 ≤ k
this : ¬IsUnit 3
hdvdp : 3 ^ 2 ∣ p ^ 2
⊢ ℕ

p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
hg : g = 3 ^ 2 * 3 ^ (k - 2)
H : 2 ≤ k
this : ¬IsUnit 3
hdvdp : 3 ^ 2 ∣ p ^ 2
⊢ ℕ
Theorem: gcd1or3
Proof state before:
case refine'_1
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
hg : g = 3 ^ 2 * 3 ^ (k - 2)
H : 2 ≤ k
this : ¬IsUnit 3
hdvdp : 3 ^ 2 ∣ p ^ 2
⊢ 3 ^ 2 ∣ ↑(?refine'_2.gcd (p ^ 2 + 3 * q ^ ?m.72510))

case refine'_2
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
hg : g = 3 ^ 2 * 3 ^ (k - 2)
H : 2 ≤ k
this : ¬IsUnit 3
hdvdp : 3 ^ 2 ∣ p ^ 2
⊢ ℤ

p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
hg : g = 3 ^ 2 * 3 ^ (k - 2)
H : 2 ≤ k
this : ¬IsUnit 3
hdvdp : 3 ^ 2 ∣ p ^ 2
⊢ ℕ

p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
hg : g = 3 ^ 2 * 3 ^ (k - 2)
H : 2 ≤ k
this : ¬IsUnit 3
hdvdp : 3 ^ 2 ∣ p ^ 2
⊢ ℕ

p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
hg : g = 3 ^ 2 * 3 ^ (k - 2)
H : 2 ≤ k
this : ¬IsUnit 3
hdvdp : 3 ^ 2 ∣ p ^ 2
⊢ ℕ

p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
hg : g = 3 ^ 2 * 3 ^ (k - 2)
H : 2 ≤ k
this : ¬IsUnit 3
hdvdp : 3 ^ 2 ∣ p ^ 2
⊢ ℕ
Tactic:
rw [← hg', hg, Int.ofNat_mul]
Proof state after:
case refine'_1
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
hg : g = 3 ^ 2 * 3 ^ (k - 2)
H : 2 ≤ k
this : ¬IsUnit 3
hdvdp : 3 ^ 2 ∣ p ^ 2
⊢ 3 ^ 2 ∣ ↑(3 ^ 2) * ↑(3 ^ (k - 2))
Theorem: gcd1or3
Proof state before:
case refine'_1
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
g : ℕ := (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
hg' : g = (2 * p).gcd (p ^ 2 + 3 * q ^ 2)
basic : ∀ (f : ℕ), f.Prime → ↑f ∣ 2 * p → ↑f ∣ p ^ 2 + 3 * q ^ 2 → f = 3
k : ℕ := g.factors.length
hg : g = 3 ^ 2 * 3 ^ (k - 2)
H : 2 ≤ k
this : ¬IsUnit 3
hdvdp : 3 ^ 2 ∣ p ^ 2
⊢ 3 ^ 2 ∣ ↑(3 ^ 2) * ↑(3 ^ (k - 2))
Tactic:
apply dvd_mul_right
Proof state after:
no goals
Theorem: obscure'
Proof state before:
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
hcube : ∃ r, p ^ 2 + 3 * q ^ 2 = r ^ 3
⊢ ∃ a b, p = a * (a - 3 * b) * (a + 3 * b) ∧ q = 3 * b * (a - b) * (a + b) ∧ IsCoprime a b ∧ (Even a ↔ ¬Even b)
Tactic:
obtain ⟨u, hu⟩ := hcube
Proof state after:
case intro
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
⊢ ∃ a b, p = a * (a - 3 * b) * (a + 3 * b) ∧ q = 3 * b * (a - b) * (a + b) ∧ IsCoprime a b ∧ (Even a ↔ ¬Even b)
Theorem: obscure'
Proof state before:
case intro
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
⊢ ∃ a b, p = a * (a - 3 * b) * (a + 3 * b) ∧ q = 3 * b * (a - b) * (a + b) ∧ IsCoprime a b ∧ (Even a ↔ ¬Even b)
Tactic:
obtain ⟨a, b, hp', hq'⟩ := step6 p q u hcoprime hu.symm
Proof state after:
case intro.intro.intro.intro
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
⊢ ∃ a b, p = a * (a - 3 * b) * (a + 3 * b) ∧ q = 3 * b * (a - b) * (a + b) ∧ IsCoprime a b ∧ (Even a ↔ ¬Even b)
Theorem: obscure'
Proof state before:
case intro.intro.intro.intro
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
⊢ ∃ a b, p = a * (a - 3 * b) * (a + 3 * b) ∧ q = 3 * b * (a - b) * (a + b) ∧ IsCoprime a b ∧ (Even a ↔ ¬Even b)
Tactic:
refine' ⟨a, b, _, _, _, _⟩
Proof state after:
case intro.intro.intro.intro.refine'_1
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
⊢ p = a * (a - 3 * b) * (a + 3 * b)

case intro.intro.intro.intro.refine'_2
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
⊢ q = 3 * b * (a - b) * (a + b)

case intro.intro.intro.intro.refine'_3
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
⊢ IsCoprime a b

case intro.intro.intro.intro.refine'_4
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
⊢ Even a ↔ ¬Even b
Theorem: obscure'
Proof state before:
case intro.intro.intro.intro.refine'_1
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
⊢ p = a * (a - 3 * b) * (a + 3 * b)
Tactic:
rw [hp']
Proof state after:
case intro.intro.intro.intro.refine'_1
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
⊢ a ^ 3 - 9 * a * b ^ 2 = a * (a - 3 * b) * (a + 3 * b)
Theorem: obscure'
Proof state before:
case intro.intro.intro.intro.refine'_1
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
⊢ a ^ 3 - 9 * a * b ^ 2 = a * (a - 3 * b) * (a + 3 * b)
Tactic:
ring
Proof state after:
no goals