IMO 2024 P2

Determine all pairs (a, b) of positive integers for which there exist positive integers g and N such that \operatorname{gcd}(a^n + b, b^n + a) = g holds for all integers n \ge N. (Note that \operatorname{gcd}(x, y) denotes the greatest common divisor of integers x and y.)

Solution: a = 1 and b = 1

open scoped Nat

theorem imo_2024_p2 :
    {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, Nat.gcd (a ^ n + b) (b ^ n + a) = g}
    = {(1, 1)} := by⊢ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g} = {(1, 1)}
  induction(10)+2zero
⊢ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g} = {(1, 1)}
succ
n✝ : ℕ
a✝ : {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g} = {(1, 1)}
⊢ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g} = {(1, 1)}
  ·zero
⊢ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g} = {(1, 1)} use Set.eq_singleton_iff_unique_mem.2 ⟨?_,λb g=>by_contra$ g.2.2.rec λY S i=>S.rec λL D=>?_⟩zero.refine_1
⊢ (1, 1) ∈ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g}
zero.refine_2
b : ℕ × ℕ
g : b ∈ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g}
Y : ℕ
S : ∃ N, 0 < Y ∧ 0 < N ∧ ∀ n ≥ N, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
L : ℕ
D : 0 < Y ∧ 0 < L ∧ ∀ n ≥ L, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
⊢ False
    ·zero.refine_1
⊢ (1, 1) ∈ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g} -- (1, 1) satisfies the condition with g = 2, N = 3.
      exact⟨by⊢ 0 < 1 leftAll goals completed! 🐙,by⊢ 0 < 1 leftAll goals completed! 🐙,2,3,by⊢ 0 < 2 ∧ 0 < 3 ∧ ∀ n ≥ 3, (1 ^ n + 1).gcd (1 ^ n + 1) = 2 simp_allAll goals completed! 🐙⟩
    -- We claim that this is the only solution.
    -- The agent wastes the next 16 lines proving then discarding a lemma.
    have:b.1+b.2∣Y:=?_zero.refine_2.refine_2
b : ℕ × ℕ
g : b ∈ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g}
Y : ℕ
S : ∃ N, 0 < Y ∧ 0 < N ∧ ∀ n ≥ N, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
L : ℕ
D : 0 < Y ∧ 0 < L ∧ ∀ n ≥ L, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
this : b.1 + b.2 ∣ Y
⊢ False
zero.refine_2.refine_1
b : ℕ × ℕ
g : b ∈ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g}
Y : ℕ
S : ∃ N, 0 < Y ∧ 0 < N ∧ ∀ n ≥ N, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
L : ℕ
D : 0 < Y ∧ 0 < L ∧ ∀ n ≥ L, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
⊢ b.1 + b.2 ∣ Y
    ·zero.refine_2.refine_2
b : ℕ × ℕ
g : b ∈ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g}
Y : ℕ
S : ∃ N, 0 < Y ∧ 0 < N ∧ ∀ n ≥ N, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
L : ℕ
D : 0 < Y ∧ 0 < L ∧ ∀ n ≥ L, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
this : b.1 + b.2 ∣ Y
⊢ False suffices: b.1= b.2zero.refine_2.refine_2
b : ℕ × ℕ
g : b ∈ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g}
Y : ℕ
S : ∃ N, 0 < Y ∧ 0 < N ∧ ∀ n ≥ N, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
L : ℕ
D : 0 < Y ∧ 0 < L ∧ ∀ n ≥ L, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
this✝ : b.1 + b.2 ∣ Y
this : b.1 = b.2
⊢ False
this
b : ℕ × ℕ
g : b ∈ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g}
Y : ℕ
S : ∃ N, 0 < Y ∧ 0 < N ∧ ∀ n ≥ N, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
L : ℕ
D : 0 < Y ∧ 0 < L ∧ ∀ n ≥ L, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
this : b.1 + b.2 ∣ Y
⊢ b.1 = b.2
      ·zero.refine_2.refine_2
b : ℕ × ℕ
g : b ∈ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g}
Y : ℕ
S : ∃ N, 0 < Y ∧ 0 < N ∧ ∀ n ≥ N, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
L : ℕ
D : 0 < Y ∧ 0 < L ∧ ∀ n ≥ L, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
this✝ : b.1 + b.2 ∣ Y
this : b.1 = b.2
⊢ False norm_num[b.ext_iff,<-D.2.2 L,this]at*zero.refine_2.refine_2
b : ℕ × ℕ
Y : ℕ
L : ℕ
g : 0 < b.2 ∧ ∃ g, 0 < g ∧ ∃ x, 0 < x ∧ ∀ (n : ℕ), x ≤ n → b.2 ^ n + b.2 = g
S : (0 < b.2 ^ L ∨ 0 < b.2) ∧ ∃ x, 0 < x ∧ ∀ (n : ℕ), x ≤ n → b.2 ^ n = b.2 ^ L
i : ¬b.2 = 1
D : (0 < b.2 ^ L ∨ 0 < b.2) ∧ 0 < L ∧ ∀ (n : ℕ), L ≤ n → b.2 ^ n = b.2 ^ L
this✝ : b.2 + b.2 ∣ b.2 ^ L + b.2
this : True
⊢ False
        use(`pow_lt_pow` has been deprecated, use `pow_lt_pow_right` insteadpow_lt_pow (g.1.nat_succ_le.lt_of_ne' i) (byb : ℕ × ℕ
Y : ℕ
L : ℕ
g : 0 < b.2 ∧ ∃ g, 0 < g ∧ ∃ x, 0 < x ∧ ∀ (n : ℕ), x ≤ n → b.2 ^ n + b.2 = g
S : (0 < b.2 ^ L ∨ 0 < b.2) ∧ ∃ x, 0 < x ∧ ∀ (n : ℕ), x ≤ n → b.2 ^ n = b.2 ^ L
i : ¬b.2 = 1
D : (0 < b.2 ^ L ∨ 0 < b.2) ∧ 0 < L ∧ ∀ (n : ℕ), L ≤ n → b.2 ^ n = b.2 ^ L
this✝ : b.2 + b.2 ∣ b.2 ^ L + b.2
this : True
⊢ L < L.succ leftAll goals completed! 🐙)).ne' (D.2.2 _ L.le_succ)
      suffices:b.1+b.2∣b.fst^ (2 *L) +b.2 ∧(b).fst +(b).snd ∣ b.snd^ (2 *L)+b.1this
b : ℕ × ℕ
g : b ∈ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g}
Y : ℕ
S : ∃ N, 0 < Y ∧ 0 < N ∧ ∀ n ≥ N, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
L : ℕ
D : 0 < Y ∧ 0 < L ∧ ∀ n ≥ L, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
this✝ : b.1 + b.2 ∣ Y
this : b.1 + b.2 ∣ b.1 ^ (2 * L) + b.2 ∧ b.1 + b.2 ∣ b.2 ^ (2 * L) + b.1
⊢ b.1 = b.2
this
b : ℕ × ℕ
g : b ∈ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g}
Y : ℕ
S : ∃ N, 0 < Y ∧ 0 < N ∧ ∀ n ≥ N, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
L : ℕ
D : 0 < Y ∧ 0 < L ∧ ∀ n ≥ L, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
this : b.1 + b.2 ∣ Y
⊢ b.1 + b.2 ∣ b.1 ^ (2 * L) + b.2 ∧ b.1 + b.2 ∣ b.2 ^ (2 * L) + b.1
      ·this
b : ℕ × ℕ
g : b ∈ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g}
Y : ℕ
S : ∃ N, 0 < Y ∧ 0 < N ∧ ∀ n ≥ N, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
L : ℕ
D : 0 < Y ∧ 0 < L ∧ ∀ n ≥ L, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
this✝ : b.1 + b.2 ∣ Y
this : b.1 + b.2 ∣ b.1 ^ (2 * L) + b.2 ∧ b.1 + b.2 ∣ b.2 ^ (2 * L) + b.1
⊢ b.1 = b.2 suffices:b.1^2%(b.1+b.2)=b.2^2%(b.1+b.snd)this
b : ℕ × ℕ
g : b ∈ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g}
Y : ℕ
S : ∃ N, 0 < Y ∧ 0 < N ∧ ∀ n ≥ N, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
L : ℕ
D : 0 < Y ∧ 0 < L ∧ ∀ n ≥ L, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
this✝¹ : b.1 + b.2 ∣ Y
this✝ : b.1 + b.2 ∣ b.1 ^ (2 * L) + b.2 ∧ b.1 + b.2 ∣ b.2 ^ (2 * L) + b.1
this : b.1 ^ 2 % (b.1 + b.2) = b.2 ^ 2 % (b.1 + b.2)
⊢ b.1 = b.2
this
b : ℕ × ℕ
g : b ∈ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g}
Y : ℕ
S : ∃ N, 0 < Y ∧ 0 < N ∧ ∀ n ≥ N, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
L : ℕ
D : 0 < Y ∧ 0 < L ∧ ∀ n ≥ L, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
this✝ : b.1 + b.2 ∣ Y
this : b.1 + b.2 ∣ b.1 ^ (2 * L) + b.2 ∧ b.1 + b.2 ∣ b.2 ^ (2 * L) + b.1
⊢ b.1 ^ 2 % (b.1 + b.2) = b.2 ^ 2 % (b.1 + b.2)
        ·this
b : ℕ × ℕ
g : b ∈ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g}
Y : ℕ
S : ∃ N, 0 < Y ∧ 0 < N ∧ ∀ n ≥ N, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
L : ℕ
D : 0 < Y ∧ 0 < L ∧ ∀ n ≥ L, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
this✝¹ : b.1 + b.2 ∣ Y
this✝ : b.1 + b.2 ∣ b.1 ^ (2 * L) + b.2 ∧ b.1 + b.2 ∣ b.2 ^ (2 * L) + b.1
this : b.1 ^ 2 % (b.1 + b.2) = b.2 ^ 2 % (b.1 + b.2)
⊢ b.1 = b.2 norm_num[Nat.add_mod,pow_mul,this,Nat.dvd_iff_mod_eq_zero,Nat.pow_mod]at*this
b : ℕ × ℕ
Y : ℕ
i : ¬b = (1, 1)
L : ℕ
g : 0 < b.1 ∧ 0 < b.2 ∧ ∃ g, 0 < g ∧ ∃ x, 0 < x ∧ ∀ (n : ℕ), x ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = g
S : 0 < Y ∧ ∃ x, 0 < x ∧ ∀ (n : ℕ), x ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
D : 0 < Y ∧ 0 < L ∧ ∀ (n : ℕ), L ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
this✝¹ : Y % (b.1 + b.2) = 0
this✝ : (((b.2 % (b.1 + b.2) % (b.1 + b.2)) ^ 2 % (b.1 + b.2) % (b.1 + b.2)) ^ L % (b.1 + b.2) % (b.1 + b.2) +
        b.2 % (b.1 + b.2) % (b.1 + b.2)) %
      (b.1 + b.2) =
    0 ∧
  (((b.2 % (b.1 + b.2) % (b.1 + b.2)) ^ 2 % (b.1 + b.2) % (b.1 + b.2)) ^ L % (b.1 + b.2) % (b.1 + b.2) +
        b.1 % (b.1 + b.2) % (b.1 + b.2)) %
      (b.1 + b.2) =
    0
this : True
⊢ b.1 = b.2
          norm_num[add_comm,b.ext_iff,sq _,←Nat.pow_mod,←Nat.dvd_iff_mod_eq_zero]at*this
b : ℕ × ℕ
Y : ℕ
L : ℕ
i : b.1 = 1 → ¬b.2 = 1
g : 0 < b.1 ∧ 0 < b.2 ∧ ∃ g, 0 < g ∧ ∃ x, 0 < x ∧ ∀ (n : ℕ), x ≤ n → (b.2 + b.1 ^ n).gcd (b.1 + b.2 ^ n) = g
S : 0 < Y ∧ ∃ x, 0 < x ∧ ∀ (n : ℕ), x ≤ n → (b.2 + b.1 ^ n).gcd (b.1 + b.2 ^ n) = Y
D : 0 < Y ∧ 0 < L ∧ ∀ (n : ℕ), L ≤ n → (b.2 + b.1 ^ n).gcd (b.1 + b.2 ^ n) = Y
this✝¹ : b.1 + b.2 ∣ Y
this✝ : b.1 + b.2 ∣ b.2 + (b.2 * b.2) ^ L ∧ b.1 + b.2 ∣ b.1 + (b.2 * b.2) ^ L
this : True
⊢ b.1 = b.2
          zify at*this
b : ℕ × ℕ
Y : ℕ
L : ℕ
this✝¹ : True
i : ↑b.1 = 1 → ¬↑b.2 = 1
g : 0 < ↑b.1 ∧ 0 < ↑b.2 ∧ ∃ g, 0 < ↑g ∧ ∃ x, 0 < ↑x ∧ ∀ (n : ℕ), ↑x ≤ ↑n → ↑((b.2 + b.1 ^ n).gcd (b.1 + b.2 ^ n)) = ↑g
S : 0 < ↑Y ∧ ∃ x, 0 < ↑x ∧ ∀ (n : ℕ), ↑x ≤ ↑n → ↑((b.2 + b.1 ^ n).gcd (b.1 + b.2 ^ n)) = ↑Y
D : 0 < ↑Y ∧ 0 < ↑L ∧ ∀ (n : ℕ), ↑L ≤ ↑n → ↑((b.2 + b.1 ^ n).gcd (b.1 + b.2 ^ n)) = ↑Y
this✝ : ↑b.1 + ↑b.2 ∣ ↑Y
this : ↑b.1 + ↑b.2 ∣ ↑b.2 + (↑b.2 * ↑b.2) ^ L ∧ ↑b.1 + ↑b.2 ∣ ↑b.1 + (↑b.2 * ↑b.2) ^ L
⊢ ↑b.1 = ↑b.2
          cases this.1.sub this.2with|_ Z=>this.intro
b : ℕ × ℕ
Y : ℕ
L : ℕ
this✝¹ : True
i : ↑b.1 = 1 → ¬↑b.2 = 1
g : 0 < ↑b.1 ∧ 0 < ↑b.2 ∧ ∃ g, 0 < ↑g ∧ ∃ x, 0 < ↑x ∧ ∀ (n : ℕ), ↑x ≤ ↑n → ↑((b.2 + b.1 ^ n).gcd (b.1 + b.2 ^ n)) = ↑g
S : 0 < ↑Y ∧ ∃ x, 0 < ↑x ∧ ∀ (n : ℕ), ↑x ≤ ↑n → ↑((b.2 + b.1 ^ n).gcd (b.1 + b.2 ^ n)) = ↑Y
D : 0 < ↑Y ∧ 0 < ↑L ∧ ∀ (n : ℕ), ↑L ≤ ↑n → ↑((b.2 + b.1 ^ n).gcd (b.1 + b.2 ^ n)) = ↑Y
this✝ : ↑b.1 + ↑b.2 ∣ ↑Y
this : ↑b.1 + ↑b.2 ∣ ↑b.2 + (↑b.2 * ↑b.2) ^ L ∧ ↑b.1 + ↑b.2 ∣ ↑b.1 + (↑b.2 * ↑b.2) ^ L
Z : ℤ
h✝ : ↑b.2 + (↑b.2 * ↑b.2) ^ L - (↑b.1 + (↑b.2 * ↑b.2) ^ L) = (↑b.1 + ↑b.2) * Z
⊢ ↑b.1 = ↑b.2 nlinarith [ (byb : ℕ × ℕ
Y : ℕ
L : ℕ
this✝¹ : True
i : ↑b.1 = 1 → ¬↑b.2 = 1
g : 0 < ↑b.1 ∧ 0 < ↑b.2 ∧ ∃ g, 0 < ↑g ∧ ∃ x, 0 < ↑x ∧ ∀ (n : ℕ), ↑x ≤ ↑n → ↑((b.2 + b.1 ^ n).gcd (b.1 + b.2 ^ n)) = ↑g
S : 0 < ↑Y ∧ ∃ x, 0 < ↑x ∧ ∀ (n : ℕ), ↑x ≤ ↑n → ↑((b.2 + b.1 ^ n).gcd (b.1 + b.2 ^ n)) = ↑Y
D : 0 < ↑Y ∧ 0 < ↑L ∧ ∀ (n : ℕ), ↑L ≤ ↑n → ↑((b.2 + b.1 ^ n).gcd (b.1 + b.2 ^ n)) = ↑Y
this✝ : ↑b.1 + ↑b.2 ∣ ↑Y
this : ↑b.1 + ↑b.2 ∣ ↑b.2 + (↑b.2 * ↑b.2) ^ L ∧ ↑b.1 + ↑b.2 ∣ ↑b.1 + (↑b.2 * ↑b.2) ^ L
Z : ℤ
h✝ : ↑b.2 + (↑b.2 * ↑b.2) ^ L - (↑b.1 + (↑b.2 * ↑b.2) ^ L) = (↑b.1 + ↑b.2) * Z
⊢ Z = 0 (nlinarithAll goals completed! 🐙): Z=0 )]
        apply@Nat.modEq_of_dvdthis.a
b : ℕ × ℕ
g : b ∈ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g}
Y : ℕ
S : ∃ N, 0 < Y ∧ 0 < N ∧ ∀ n ≥ N, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
L : ℕ
D : 0 < Y ∧ 0 < L ∧ ∀ n ≥ L, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
this✝ : b.1 + b.2 ∣ Y
this : b.1 + b.2 ∣ b.1 ^ (2 * L) + b.2 ∧ b.1 + b.2 ∣ b.2 ^ (2 * L) + b.1
⊢ ↑(b.1 + b.2) ∣ ↑(b.2 ^ 2) - ↑(b.1 ^ 2)
        use(b.snd)-b.fst , (byb : ℕ × ℕ
g : b ∈ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g}
Y : ℕ
S : ∃ N, 0 < Y ∧ 0 < N ∧ ∀ n ≥ N, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
L : ℕ
D : 0 < Y ∧ 0 < L ∧ ∀ n ≥ L, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
this✝ : b.1 + b.2 ∣ Y
this : b.1 + b.2 ∣ b.1 ^ (2 * L) + b.2 ∧ b.1 + b.2 ∣ b.2 ^ (2 * L) + b.1
⊢ ↑b.2 ^ 2 - ↑b.1 ^ 2 = (↑b.1 + ↑b.2) * (↑b.2 - ↑b.1)·b : ℕ × ℕ
g : b ∈ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g}
Y : ℕ
S : ∃ N, 0 < Y ∧ 0 < N ∧ ∀ n ≥ N, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
L : ℕ
D : 0 < Y ∧ 0 < L ∧ ∀ n ≥ L, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
this✝ : b.1 + b.2 ∣ Y
this : b.1 + b.2 ∣ b.1 ^ (2 * L) + b.2 ∧ b.1 + b.2 ∣ b.2 ^ (2 * L) + b.1
⊢ ↑b.2 ^ 2 - ↑b.1 ^ 2 = (↑b.1 + ↑b.2) * (↑b.2 - ↑b.1)ringAll goals completed! 🐙: ( (b.snd) : ℤ)^2-b.fst^2=(b.fst+(b).2) * _)
      norm_num[(2).le_mul_of_pos_left,Nat.gcd_dvd,← D.2.2 (2 *L), this.trans, (D.right.1 :_)]All goals completed! 🐙
    suffices:b.1+b.2∣b.1^(2*L)+b.2 ∧b.1+b.2 ∣b.snd^ (2 *L) +b.1zero.refine_2.refine_1
b : ℕ × ℕ
g : b ∈ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g}
Y : ℕ
S : ∃ N, 0 < Y ∧ 0 < N ∧ ∀ n ≥ N, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
L : ℕ
D : 0 < Y ∧ 0 < L ∧ ∀ n ≥ L, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
this : b.1 + b.2 ∣ b.1 ^ (2 * L) + b.2 ∧ b.1 + b.2 ∣ b.2 ^ (2 * L) + b.1
⊢ b.1 + b.2 ∣ Y
this
b : ℕ × ℕ
g : b ∈ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g}
Y : ℕ
S : ∃ N, 0 < Y ∧ 0 < N ∧ ∀ n ≥ N, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
L : ℕ
D : 0 < Y ∧ 0 < L ∧ ∀ n ≥ L, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
⊢ b.1 + b.2 ∣ b.1 ^ (2 * L) + b.2 ∧ b.1 + b.2 ∣ b.2 ^ (2 * L) + b.1
    ·zero.refine_2.refine_1
b : ℕ × ℕ
g : b ∈ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g}
Y : ℕ
S : ∃ N, 0 < Y ∧ 0 < N ∧ ∀ n ≥ N, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
L : ℕ
D : 0 < Y ∧ 0 < L ∧ ∀ n ≥ L, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
this : b.1 + b.2 ∣ b.1 ^ (2 * L) + b.2 ∧ b.1 + b.2 ∣ b.2 ^ (2 * L) + b.1
⊢ b.1 + b.2 ∣ Y exact D.2.2 (2 *(L )) (le_mul_of_one_le_left' (byb : ℕ × ℕ
g : b ∈ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g}
Y : ℕ
S : ∃ N, 0 < Y ∧ 0 < N ∧ ∀ n ≥ N, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
L : ℕ
D : 0 < Y ∧ 0 < L ∧ ∀ n ≥ L, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
this : b.1 + b.2 ∣ b.1 ^ (2 * L) + b.2 ∧ b.1 + b.2 ∣ b.2 ^ (2 * L) + b.1
⊢ 1 ≤ 2 decideAll goals completed! 🐙 ) )▸dvd_gcd (this.left) (this).2
    exfalsothis
b : ℕ × ℕ
g : b ∈ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g}
Y : ℕ
S : ∃ N, 0 < Y ∧ 0 < N ∧ ∀ n ≥ N, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
L : ℕ
D : 0 < Y ∧ 0 < L ∧ ∀ n ≥ L, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
⊢ False
    -- First, assume ab + 1 | g.
    suffices:b.1*b.2+1∣Ythis
b : ℕ × ℕ
g : b ∈ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g}
Y : ℕ
S : ∃ N, 0 < Y ∧ 0 < N ∧ ∀ n ≥ N, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
L : ℕ
D : 0 < Y ∧ 0 < L ∧ ∀ n ≥ L, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
this : b.1 * b.2 + 1 ∣ Y
⊢ False
this
b : ℕ × ℕ
g : b ∈ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g}
Y : ℕ
S : ∃ N, 0 < Y ∧ 0 < N ∧ ∀ n ≥ N, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
L : ℕ
D : 0 < Y ∧ 0 < L ∧ ∀ n ≥ L, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
⊢ b.1 * b.2 + 1 ∣ Y
    ·this
b : ℕ × ℕ
g : b ∈ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g}
Y : ℕ
S : ∃ N, 0 < Y ∧ 0 < N ∧ ∀ n ≥ N, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
L : ℕ
D : 0 < Y ∧ 0 < L ∧ ∀ n ≥ L, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
this : b.1 * b.2 + 1 ∣ Y
⊢ False suffices:b.1^φ (b.1*b.2+1)%(b.1*b.2+1)=1%(b.1*b.2+1) ∧b.2^ φ (b.1* b.snd+1)%((b).1 * ↑(b.snd)+1)= 1% (b.1*b.snd + 1)this
b : ℕ × ℕ
g : b ∈ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g}
Y : ℕ
S : ∃ N, 0 < Y ∧ 0 < N ∧ ∀ n ≥ N, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
L : ℕ
D : 0 < Y ∧ 0 < L ∧ ∀ n ≥ L, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
this✝ : b.1 * b.2 + 1 ∣ Y
this : b.1 ^ φ (b.1 * b.2 + 1) % (b.1 * b.2 + 1) = 1 % (b.1 * b.2 + 1) ∧
  b.2 ^ φ (b.1 * b.2 + 1) % (b.1 * b.2 + 1) = 1 % (b.1 * b.2 + 1)
⊢ False
this
b : ℕ × ℕ
g : b ∈ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g}
Y : ℕ
S : ∃ N, 0 < Y ∧ 0 < N ∧ ∀ n ≥ N, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
L : ℕ
D : 0 < Y ∧ 0 < L ∧ ∀ n ≥ L, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
this : b.1 * b.2 + 1 ∣ Y
⊢ b.1 ^ φ (b.1 * b.2 + 1) % (b.1 * b.2 + 1) = 1 % (b.1 * b.2 + 1) ∧
  b.2 ^ φ (b.1 * b.2 + 1) % (b.1 * b.2 + 1) = 1 % (b.1 * b.2 + 1)
      ·this
b : ℕ × ℕ
g : b ∈ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g}
Y : ℕ
S : ∃ N, 0 < Y ∧ 0 < N ∧ ∀ n ≥ N, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
L : ℕ
D : 0 < Y ∧ 0 < L ∧ ∀ n ≥ L, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
this✝ : b.1 * b.2 + 1 ∣ Y
this : b.1 ^ φ (b.1 * b.2 + 1) % (b.1 * b.2 + 1) = 1 % (b.1 * b.2 + 1) ∧
  b.2 ^ φ (b.1 * b.2 + 1) % (b.1 * b.2 + 1) = 1 % (b.1 * b.2 + 1)
⊢ False /- Then ab + 1 | a ^ (Nφ(ab + 1)) + b and
          ab + 1 | b ^ (Nφ(ab + 1)) + a. -/
        absurd D.2.2 (φ (b.1*b.2+1)*L) (byb : ℕ × ℕ
g : b ∈ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g}
Y : ℕ
S : ∃ N, 0 < Y ∧ 0 < N ∧ ∀ n ≥ N, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
L : ℕ
D : 0 < Y ∧ 0 < L ∧ ∀ n ≥ L, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
this✝ : b.1 * b.2 + 1 ∣ Y
this : b.1 ^ φ (b.1 * b.2 + 1) % (b.1 * b.2 + 1) = 1 % (b.1 * b.2 + 1) ∧
  b.2 ^ φ (b.1 * b.2 + 1) % (b.1 * b.2 + 1) = 1 % (b.1 * b.2 + 1)
⊢ φ (b.1 * b.2 + 1) * L ≥ L nlinarith [((b.fst *b.2+1).totient_pos).2 ↑ Fin.size_pos']All goals completed! 🐙)
        apply mt (.▸Nat.gcd_dvd _ _)this
b : ℕ × ℕ
g : b ∈ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g}
Y : ℕ
S : ∃ N, 0 < Y ∧ 0 < N ∧ ∀ n ≥ N, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
L : ℕ
D : 0 < Y ∧ 0 < L ∧ ∀ n ≥ L, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
this✝ : b.1 * b.2 + 1 ∣ Y
this : b.1 ^ φ (b.1 * b.2 + 1) % (b.1 * b.2 + 1) = 1 % (b.1 * b.2 + 1) ∧
  b.2 ^ φ (b.1 * b.2 + 1) % (b.1 * b.2 + 1) = 1 % (b.1 * b.2 + 1)
⊢ ¬(Y ∣ b.1 ^ (φ (b.1 * b.2 + 1) * L) + b.2 ∧ Y ∣ b.2 ^ (φ (b.1 * b.2 + 1) * L) + b.1)
        useλH=>absurd (‹_∣Y›.trans H.1) (λv=>absurd (‹_∣Y›.trans H.2) ? _)this
b : ℕ × ℕ
g : b ∈ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g}
Y : ℕ
S : ∃ N, 0 < Y ∧ 0 < N ∧ ∀ n ≥ N, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
L : ℕ
D : 0 < Y ∧ 0 < L ∧ ∀ n ≥ L, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
this✝ : b.1 * b.2 + 1 ∣ Y
this : b.1 ^ φ (b.1 * b.2 + 1) % (b.1 * b.2 + 1) = 1 % (b.1 * b.2 + 1) ∧
  b.2 ^ φ (b.1 * b.2 + 1) % (b.1 * b.2 + 1) = 1 % (b.1 * b.2 + 1)
H : Y ∣ b.1 ^ (φ (b.1 * b.2 + 1) * L) + b.2 ∧ Y ∣ b.2 ^ (φ (b.1 * b.2 + 1) * L) + b.1
v : b.1 * b.2 + 1 ∣ b.1 ^ (φ (b.1 * b.2 + 1) * L) + b.2
⊢ ¬b.1 * b.2 + 1 ∣ b.2 ^ (φ (b.1 * b.2 + 1) * L) + b.1
        norm_num[pow_mul,b.ext_iff,(1).mod_eq_of_lt,g.symm,this,Nat.add_mod,Nat.dvd_iff_mod_eq_zero,Nat.pow_mod]at(i)v⊢this
b : ℕ × ℕ
g : b ∈ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g}
Y : ℕ
S : ∃ N, 0 < Y ∧ 0 < N ∧ ∀ n ≥ N, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
L : ℕ
D : 0 < Y ∧ 0 < L ∧ ∀ n ≥ L, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
this✝ : b.1 * b.2 + 1 ∣ Y
this : b.1 ^ φ (b.1 * b.2 + 1) % (b.1 * b.2 + 1) = 1 % (b.1 * b.2 + 1) ∧
  b.2 ^ φ (b.1 * b.2 + 1) % (b.1 * b.2 + 1) = 1 % (b.1 * b.2 + 1)
H : Y ∣ b.1 ^ (φ (b.1 * b.2 + 1) * L) + b.2 ∧ Y ∣ b.2 ^ (φ (b.1 * b.2 + 1) * L) + b.1
i : b.1 = 1 → ¬b.2 = 1
v : (1 % (b.1 * b.2 + 1) + b.2 % (b.1 * b.2 + 1) % (b.1 * b.2 + 1)) % (b.1 * b.2 + 1) = 0
⊢ ¬(1 % (b.1 * b.2 + 1) + b.1 % (b.1 * b.2 + 1) % (b.1 * b.2 + 1)) % (b.1 * b.2 + 1) = 0
        /- By Euler's Theorem, a ^ (Nφ(ab + 1)) ≡ 1 (mod ab + 1),
          so ab + 1 | b + 1 and ab + 1 | a + 1. -/
        norm_num[add_comm,pow_mul,<-Nat.dvd_iff_mod_eq_zero]at*this
b : ℕ × ℕ
Y : ℕ
L : ℕ
this✝ : b.1 * b.2 + 1 ∣ Y
this : b.1 ^ φ (b.1 * b.2 + 1) % (b.1 * b.2 + 1) = 1 % (b.1 * b.2 + 1) ∧
  b.2 ^ φ (b.1 * b.2 + 1) % (b.1 * b.2 + 1) = 1 % (b.1 * b.2 + 1)
i : b.1 = 1 → ¬b.2 = 1
g : 0 < b.1 ∧ 0 < b.2 ∧ ∃ g, 0 < g ∧ ∃ x, 0 < x ∧ ∀ (n : ℕ), x ≤ n → (b.2 + b.1 ^ n).gcd (b.1 + b.2 ^ n) = g
S : 0 < Y ∧ ∃ x, 0 < x ∧ ∀ (n : ℕ), x ≤ n → (b.2 + b.1 ^ n).gcd (b.1 + b.2 ^ n) = Y
D : 0 < Y ∧ 0 < L ∧ ∀ (n : ℕ), L ≤ n → (b.2 + b.1 ^ n).gcd (b.1 + b.2 ^ n) = Y
H : Y ∣ b.2 + (b.1 ^ φ (b.1 * b.2 + 1)) ^ L ∧ Y ∣ b.1 + (b.2 ^ φ (b.1 * b.2 + 1)) ^ L
v : b.1 * b.2 + 1 ∣ b.2 + 1
⊢ ¬b.1 * b.2 + 1 ∣ b.1 + 1
        contrapose! ithis
b : ℕ × ℕ
Y : ℕ
L : ℕ
this✝ : b.1 * b.2 + 1 ∣ Y
this : b.1 ^ φ (b.1 * b.2 + 1) % (b.1 * b.2 + 1) = 1 % (b.1 * b.2 + 1) ∧
  b.2 ^ φ (b.1 * b.2 + 1) % (b.1 * b.2 + 1) = 1 % (b.1 * b.2 + 1)
g : 0 < b.1 ∧ 0 < b.2 ∧ ∃ g, 0 < g ∧ ∃ x, 0 < x ∧ ∀ (n : ℕ), x ≤ n → (b.2 + b.1 ^ n).gcd (b.1 + b.2 ^ n) = g
S : 0 < Y ∧ ∃ x, 0 < x ∧ ∀ (n : ℕ), x ≤ n → (b.2 + b.1 ^ n).gcd (b.1 + b.2 ^ n) = Y
D : 0 < Y ∧ 0 < L ∧ ∀ (n : ℕ), L ≤ n → (b.2 + b.1 ^ n).gcd (b.1 + b.2 ^ n) = Y
H : Y ∣ b.2 + (b.1 ^ φ (b.1 * b.2 + 1)) ^ L ∧ Y ∣ b.1 + (b.2 ^ φ (b.1 * b.2 + 1)) ^ L
v : b.1 * b.2 + 1 ∣ b.2 + 1
i : b.1 * b.2 + 1 ∣ b.1 + 1
⊢ b.1 = 1 ∧ b.2 = 1
        zify at*this
b : ℕ × ℕ
Y : ℕ
L : ℕ
this✝ : ↑b.1 * ↑b.2 + 1 ∣ ↑Y
this : ↑b.1 ^ φ (b.1 * b.2 + 1) % (↑b.1 * ↑b.2 + 1) = 1 % (↑b.1 * ↑b.2 + 1) ∧
  ↑b.2 ^ φ (b.1 * b.2 + 1) % (↑b.1 * ↑b.2 + 1) = 1 % (↑b.1 * ↑b.2 + 1)
g : 0 < ↑b.1 ∧ 0 < ↑b.2 ∧ ∃ g, 0 < ↑g ∧ ∃ x, 0 < ↑x ∧ ∀ (n : ℕ), ↑x ≤ ↑n → ↑((b.2 + b.1 ^ n).gcd (b.1 + b.2 ^ n)) = ↑g
S : 0 < ↑Y ∧ ∃ x, 0 < ↑x ∧ ∀ (n : ℕ), ↑x ≤ ↑n → ↑((b.2 + b.1 ^ n).gcd (b.1 + b.2 ^ n)) = ↑Y
D : 0 < ↑Y ∧ 0 < ↑L ∧ ∀ (n : ℕ), ↑L ≤ ↑n → ↑((b.2 + b.1 ^ n).gcd (b.1 + b.2 ^ n)) = ↑Y
H : ↑Y ∣ ↑b.2 + (↑b.1 ^ φ (b.1 * b.2 + 1)) ^ L ∧ ↑Y ∣ ↑b.1 + (↑b.2 ^ φ (b.1 * b.2 + 1)) ^ L
v : ↑b.1 * ↑b.2 + 1 ∣ ↑b.2 + 1
i : ↑b.1 * ↑b.2 + 1 ∣ ↑b.1 + 1
⊢ ↑b.1 = 1 ∧ ↑b.2 = 1
        /- Thus ab + 1 ≤ b + 1 and ab + 1 ≤ a + 1 which requires
          a = b = 1 as desired. -/
        repeat use byb : ℕ × ℕ
Y : ℕ
L : ℕ
this✝ : ↑b.1 * ↑b.2 + 1 ∣ ↑Y
this : ↑b.1 ^ φ (b.1 * b.2 + 1) % (↑b.1 * ↑b.2 + 1) = 1 % (↑b.1 * ↑b.2 + 1) ∧
  ↑b.2 ^ φ (b.1 * b.2 + 1) % (↑b.1 * ↑b.2 + 1) = 1 % (↑b.1 * ↑b.2 + 1)
g : 0 < ↑b.1 ∧ 0 < ↑b.2 ∧ ∃ g, 0 < ↑g ∧ ∃ x, 0 < ↑x ∧ ∀ (n : ℕ), ↑x ≤ ↑n → ↑((b.2 + b.1 ^ n).gcd (b.1 + b.2 ^ n)) = ↑g
S : 0 < ↑Y ∧ ∃ x, 0 < ↑x ∧ ∀ (n : ℕ), ↑x ≤ ↑n → ↑((b.2 + b.1 ^ n).gcd (b.1 + b.2 ^ n)) = ↑Y
D : 0 < ↑Y ∧ 0 < ↑L ∧ ∀ (n : ℕ), ↑L ≤ ↑n → ↑((b.2 + b.1 ^ n).gcd (b.1 + b.2 ^ n)) = ↑Y
H : ↑Y ∣ ↑b.2 + (↑b.1 ^ φ (b.1 * b.2 + 1)) ^ L ∧ ↑Y ∣ ↑b.1 + (↑b.2 ^ φ (b.1 * b.2 + 1)) ^ L
v : ↑b.1 * ↑b.2 + 1 ∣ ↑b.2 + 1
i : ↑b.1 * ↑b.2 + 1 ∣ ↑b.1 + 1
⊢ ↑b.2 = 1 nlinarith[Int.le_of_dvd (byb : ℕ × ℕ
Y : ℕ
L : ℕ
this✝ : ↑b.1 * ↑b.2 + 1 ∣ ↑Y
this : ↑b.1 ^ φ (b.1 * b.2 + 1) % (↑b.1 * ↑b.2 + 1) = 1 % (↑b.1 * ↑b.2 + 1) ∧
  ↑b.2 ^ φ (b.1 * b.2 + 1) % (↑b.1 * ↑b.2 + 1) = 1 % (↑b.1 * ↑b.2 + 1)
g : 0 < ↑b.1 ∧ 0 < ↑b.2 ∧ ∃ g, 0 < ↑g ∧ ∃ x, 0 < ↑x ∧ ∀ (n : ℕ), ↑x ≤ ↑n → ↑((b.2 + b.1 ^ n).gcd (b.1 + b.2 ^ n)) = ↑g
S : 0 < ↑Y ∧ ∃ x, 0 < ↑x ∧ ∀ (n : ℕ), ↑x ≤ ↑n → ↑((b.2 + b.1 ^ n).gcd (b.1 + b.2 ^ n)) = ↑Y
D : 0 < ↑Y ∧ 0 < ↑L ∧ ∀ (n : ℕ), ↑L ≤ ↑n → ↑((b.2 + b.1 ^ n).gcd (b.1 + b.2 ^ n)) = ↑Y
H : ↑Y ∣ ↑b.2 + (↑b.1 ^ φ (b.1 * b.2 + 1)) ^ L ∧ ↑Y ∣ ↑b.1 + (↑b.2 ^ φ (b.1 * b.2 + 1)) ^ L
v : ↑b.1 * ↑b.2 + 1 ∣ ↑b.2 + 1
i : ↑b.1 * ↑b.2 + 1 ∣ ↑b.1 + 1
⊢ 0 < ↑b.2 + 1 linarithAll goals completed! 🐙) v,Int.le_of_dvd (byb : ℕ × ℕ
Y : ℕ
L : ℕ
this✝ : ↑b.1 * ↑b.2 + 1 ∣ ↑Y
this : ↑b.1 ^ φ (b.1 * b.2 + 1) % (↑b.1 * ↑b.2 + 1) = 1 % (↑b.1 * ↑b.2 + 1) ∧
  ↑b.2 ^ φ (b.1 * b.2 + 1) % (↑b.1 * ↑b.2 + 1) = 1 % (↑b.1 * ↑b.2 + 1)
g : 0 < ↑b.1 ∧ 0 < ↑b.2 ∧ ∃ g, 0 < ↑g ∧ ∃ x, 0 < ↑x ∧ ∀ (n : ℕ), ↑x ≤ ↑n → ↑((b.2 + b.1 ^ n).gcd (b.1 + b.2 ^ n)) = ↑g
S : 0 < ↑Y ∧ ∃ x, 0 < ↑x ∧ ∀ (n : ℕ), ↑x ≤ ↑n → ↑((b.2 + b.1 ^ n).gcd (b.1 + b.2 ^ n)) = ↑Y
D : 0 < ↑Y ∧ 0 < ↑L ∧ ∀ (n : ℕ), ↑L ≤ ↑n → ↑((b.2 + b.1 ^ n).gcd (b.1 + b.2 ^ n)) = ↑Y
H : ↑Y ∣ ↑b.2 + (↑b.1 ^ φ (b.1 * b.2 + 1)) ^ L ∧ ↑Y ∣ ↑b.1 + (↑b.2 ^ φ (b.1 * b.2 + 1)) ^ L
v : ↑b.1 * ↑b.2 + 1 ∣ ↑b.2 + 1
i : ↑b.1 * ↑b.2 + 1 ∣ ↑b.1 + 1
⊢ 0 < ↑b.1 + 1 linarithAll goals completed! 🐙) i]
      repeat use↑(Nat.ModEq.pow_totient (byb : ℕ × ℕ
g : b ∈ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g}
Y : ℕ
S : ∃ N, 0 < Y ∧ 0 < N ∧ ∀ n ≥ N, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
L : ℕ
D : 0 < Y ∧ 0 < L ∧ ∀ n ≥ L, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
this : b.1 * b.2 + 1 ∣ Y
⊢ b.2.Coprime (b.1 * b.2 + 1) norm_numAll goals completed! 🐙))
    -- Now, we proceed to show that indeed ab + 1 | g.
    by_contra! Hthis
b : ℕ × ℕ
g : b ∈ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g}
Y : ℕ
S : ∃ N, 0 < Y ∧ 0 < N ∧ ∀ n ≥ N, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
L : ℕ
D : 0 < Y ∧ 0 < L ∧ ∀ n ≥ L, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
H : ¬b.1 * b.2 + 1 ∣ Y
⊢ False
    suffices:b.1^φ (b.1*b.2+1)%(b.1*b.2+1)=1%(b.1*b.2+1) ∧b.2^φ (b.1*b.2+1)%(b.1*b.2+1)=1%( b.fst * ↑ (b.snd)+1)this
b : ℕ × ℕ
g : b ∈ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g}
Y : ℕ
S : ∃ N, 0 < Y ∧ 0 < N ∧ ∀ n ≥ N, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
L : ℕ
D : 0 < Y ∧ 0 < L ∧ ∀ n ≥ L, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
H : ¬b.1 * b.2 + 1 ∣ Y
this : b.1 ^ φ (b.1 * b.2 + 1) % (b.1 * b.2 + 1) = 1 % (b.1 * b.2 + 1) ∧
  b.2 ^ φ (b.1 * b.2 + 1) % (b.1 * b.2 + 1) = 1 % (b.1 * b.2 + 1)
⊢ False
this
b : ℕ × ℕ
g : b ∈ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g}
Y : ℕ
S : ∃ N, 0 < Y ∧ 0 < N ∧ ∀ n ≥ N, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
L : ℕ
D : 0 < Y ∧ 0 < L ∧ ∀ n ≥ L, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
H : ¬b.1 * b.2 + 1 ∣ Y
⊢ b.1 ^ φ (b.1 * b.2 + 1) % (b.1 * b.2 + 1) = 1 % (b.1 * b.2 + 1) ∧
  b.2 ^ φ (b.1 * b.2 + 1) % (b.1 * b.2 + 1) = 1 % (b.1 * b.2 + 1)
    ·this
b : ℕ × ℕ
g : b ∈ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g}
Y : ℕ
S : ∃ N, 0 < Y ∧ 0 < N ∧ ∀ n ≥ N, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
L : ℕ
D : 0 < Y ∧ 0 < L ∧ ∀ n ≥ L, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
H : ¬b.1 * b.2 + 1 ∣ Y
this : b.1 ^ φ (b.1 * b.2 + 1) % (b.1 * b.2 + 1) = 1 % (b.1 * b.2 + 1) ∧
  b.2 ^ φ (b.1 * b.2 + 1) % (b.1 * b.2 + 1) = 1 % (b.1 * b.2 + 1)
⊢ False simp_allthis
b : ℕ × ℕ
Y : ℕ
L : ℕ
g : 0 < b.1 ∧ 0 < b.2 ∧ ∃ g, 0 < g ∧ ∃ x, 0 < x ∧ ∀ (n : ℕ), x ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = g
S : ∃ N, 0 < N ∧ ∀ (n : ℕ), N ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
D : 0 < Y ∧ 0 < L ∧ ∀ (n : ℕ), L ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
H : ¬b.1 * b.2 + 1 ∣ Y
this : b.1 ^ φ (b.1 * b.2 + 1) % (b.1 * b.2 + 1) = 1 % (b.1 * b.2 + 1) ∧
  b.2 ^ φ (b.1 * b.2 + 1) % (b.1 * b.2 + 1) = 1 % (b.1 * b.2 + 1)
⊢ False
      /- It suffices to show that
        ab + 1 | a^(φ(ab + 1)(N + 1) - 1) + b and
        ab + 1 | b^(φ(ab + 1)(N + 1) - 1) + a. -/
      suffices:b.1*b.2+1∣b.1^(φ (b.1*b.2+1)*(L+1)-1)+b.2 ∧b.1*b.2+1∣b.2^(φ (b.1* b.2+1)* (L+1)-1)+(b.fst)this
b : ℕ × ℕ
Y : ℕ
L : ℕ
g : 0 < b.1 ∧ 0 < b.2 ∧ ∃ g, 0 < g ∧ ∃ x, 0 < x ∧ ∀ (n : ℕ), x ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = g
S : ∃ N, 0 < N ∧ ∀ (n : ℕ), N ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
D : 0 < Y ∧ 0 < L ∧ ∀ (n : ℕ), L ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
H : ¬b.1 * b.2 + 1 ∣ Y
this✝ : b.1 ^ φ (b.1 * b.2 + 1) % (b.1 * b.2 + 1) = 1 % (b.1 * b.2 + 1) ∧
  b.2 ^ φ (b.1 * b.2 + 1) % (b.1 * b.2 + 1) = 1 % (b.1 * b.2 + 1)
this : b.1 * b.2 + 1 ∣ b.1 ^ (φ (b.1 * b.2 + 1) * (L + 1) - 1) + b.2 ∧
  b.1 * b.2 + 1 ∣ b.2 ^ (φ (b.1 * b.2 + 1) * (L + 1) - 1) + b.1
⊢ False
this
b : ℕ × ℕ
Y : ℕ
L : ℕ
g : 0 < b.1 ∧ 0 < b.2 ∧ ∃ g, 0 < g ∧ ∃ x, 0 < x ∧ ∀ (n : ℕ), x ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = g
S : ∃ N, 0 < N ∧ ∀ (n : ℕ), N ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
D : 0 < Y ∧ 0 < L ∧ ∀ (n : ℕ), L ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
H : ¬b.1 * b.2 + 1 ∣ Y
this : b.1 ^ φ (b.1 * b.2 + 1) % (b.1 * b.2 + 1) = 1 % (b.1 * b.2 + 1) ∧
  b.2 ^ φ (b.1 * b.2 + 1) % (b.1 * b.2 + 1) = 1 % (b.1 * b.2 + 1)
⊢ b.1 * b.2 + 1 ∣ b.1 ^ (φ (b.1 * b.2 + 1) * (L + 1) - 1) + b.2 ∧
  b.1 * b.2 + 1 ∣ b.2 ^ (φ (b.1 * b.2 + 1) * (L + 1) - 1) + b.1
      ·this
b : ℕ × ℕ
Y : ℕ
L : ℕ
g : 0 < b.1 ∧ 0 < b.2 ∧ ∃ g, 0 < g ∧ ∃ x, 0 < x ∧ ∀ (n : ℕ), x ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = g
S : ∃ N, 0 < N ∧ ∀ (n : ℕ), N ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
D : 0 < Y ∧ 0 < L ∧ ∀ (n : ℕ), L ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
H : ¬b.1 * b.2 + 1 ∣ Y
this✝ : b.1 ^ φ (b.1 * b.2 + 1) % (b.1 * b.2 + 1) = 1 % (b.1 * b.2 + 1) ∧
  b.2 ^ φ (b.1 * b.2 + 1) % (b.1 * b.2 + 1) = 1 % (b.1 * b.2 + 1)
this : b.1 * b.2 + 1 ∣ b.1 ^ (φ (b.1 * b.2 + 1) * (L + 1) - 1) + b.2 ∧
  b.1 * b.2 + 1 ∣ b.2 ^ (φ (b.1 * b.2 + 1) * (L + 1) - 1) + b.1
⊢ False use H$ D.2.2 (φ _ *(L+1)-1) (L.le_sub_of_add_le (byb : ℕ × ℕ
Y : ℕ
L : ℕ
g : 0 < b.1 ∧ 0 < b.2 ∧ ∃ g, 0 < g ∧ ∃ x, 0 < x ∧ ∀ (n : ℕ), x ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = g
S : ∃ N, 0 < N ∧ ∀ (n : ℕ), N ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
D : 0 < Y ∧ 0 < L ∧ ∀ (n : ℕ), L ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
H : ¬b.1 * b.2 + 1 ∣ Y
this✝ : b.1 ^ φ (b.1 * b.2 + 1) % (b.1 * b.2 + 1) = 1 % (b.1 * b.2 + 1) ∧
  b.2 ^ φ (b.1 * b.2 + 1) % (b.1 * b.2 + 1) = 1 % (b.1 * b.2 + 1)
this : b.1 * b.2 + 1 ∣ b.1 ^ (φ (b.1 * b.2 + 1) * (L + 1) - 1) + b.2 ∧
  b.1 * b.2 + 1 ∣ b.2 ^ (φ (b.1 * b.2 + 1) * (L + 1) - 1) + b.1
⊢ L + 1 ≤ φ (b.1 * b.2 + 1) * (L + 1) nlinarith[((b.1* b.2+1).totient_pos).2 Nat.succ_pos']All goals completed! 🐙))▸(((Nat.dvd_gcd) ( this).1)) this.right
      cases unused variable `B`
note: this linter can be disabled with `set_option linter.unusedVariables false`B:Nat.exists_eq_add_of_lt$ ((b.1*b.2+1).totient_pos).2 (byb : ℕ × ℕ
Y : ℕ
L : ℕ
g : 0 < b.1 ∧ 0 < b.2 ∧ ∃ g, 0 < g ∧ ∃ x, 0 < x ∧ ∀ (n : ℕ), x ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = g
S : ∃ N, 0 < N ∧ ∀ (n : ℕ), N ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
D : 0 < Y ∧ 0 < L ∧ ∀ (n : ℕ), L ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
H : ¬b.1 * b.2 + 1 ∣ Y
this : b.1 ^ φ (b.1 * b.2 + 1) % (b.1 * b.2 + 1) = 1 % (b.1 * b.2 + 1) ∧
  b.2 ^ φ (b.1 * b.2 + 1) % (b.1 * b.2 + 1) = 1 % (b.1 * b.2 + 1)
⊢ 0 < b.1 * b.2 + 1 continuityAll goals completed! 🐙)
      norm_num[*, g, ‹φ _ = _›, mul_add,Nat.pow_mod,(1).mod_eq_of_lt,pow_add,Nat.add_mod,pow_mul,Nat.dvd_iff_mod_eq_zero,Nat.mul_mod] at this⊢this.intro
b : ℕ × ℕ
Y : ℕ
L : ℕ
g : 0 < b.1 ∧ 0 < b.2 ∧ ∃ g, 0 < g ∧ ∃ x, 0 < x ∧ ∀ (n : ℕ), x ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = g
S : ∃ N, 0 < N ∧ ∀ (n : ℕ), N ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
D : 0 < Y ∧ 0 < L ∧ ∀ (n : ℕ), L ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
H : ¬b.1 * b.2 + 1 ∣ Y
w✝ : ℕ
h✝ : φ (b.1 * b.2 + 1) = 0 + w✝ + 1
B : ⋯ = ⋯
this : (b.1 % (b.1 * b.2 + 1) % (b.1 * b.2 + 1)) ^ w✝ % (b.1 * b.2 + 1) % (b.1 * b.2 + 1) *
        (b.1 % (b.1 * b.2 + 1) % (b.1 * b.2 + 1)) %
      (b.1 * b.2 + 1) =
    1 ∧
  (b.2 % (b.1 * b.2 + 1) % (b.1 * b.2 + 1)) ^ w✝ % (b.1 * b.2 + 1) % (b.1 * b.2 + 1) *
        (b.2 % (b.1 * b.2 + 1) % (b.1 * b.2 + 1)) %
      (b.1 * b.2 + 1) =
    1
⊢ (((b.1 % (b.1 * b.2 + 1) % (b.1 * b.2 + 1)) ^ w✝ % (b.1 * b.2 + 1) % (b.1 * b.2 + 1) *
                          (b.1 % (b.1 * b.2 + 1) % (b.1 * b.2 + 1)) %
                        (b.1 * b.2 + 1) %
                      (b.1 * b.2 + 1)) ^
                    L %
                  (b.1 * b.2 + 1) %
                (b.1 * b.2 + 1) *
              ((b.1 % (b.1 * b.2 + 1) % (b.1 * b.2 + 1)) ^ w✝ % (b.1 * b.2 + 1) % (b.1 * b.2 + 1)) %
            (b.1 * b.2 + 1) %
          (b.1 * b.2 + 1) +
        b.2 % (b.1 * b.2 + 1) % (b.1 * b.2 + 1)) %
      (b.1 * b.2 + 1) =
    0 ∧
  (((b.2 % (b.1 * b.2 + 1) % (b.1 * b.2 + 1)) ^ w✝ % (b.1 * b.2 + 1) % (b.1 * b.2 + 1) *
                          (b.2 % (b.1 * b.2 + 1) % (b.1 * b.2 + 1)) %
                        (b.1 * b.2 + 1) %
                      (b.1 * b.2 + 1)) ^
                    L %
                  (b.1 * b.2 + 1) %
                (b.1 * b.2 + 1) *
              ((b.2 % (b.1 * b.2 + 1) % (b.1 * b.2 + 1)) ^ w✝ % (b.1 * b.2 + 1) % (b.1 * b.2 + 1)) %
            (b.1 * b.2 + 1) %
          (b.1 * b.2 + 1) +
        b.1 % (b.1 * b.2 + 1) % (b.1 * b.2 + 1)) %
      (b.1 * b.2 + 1) =
    0
      simp_allthis.intro
b : ℕ × ℕ
Y : ℕ
L : ℕ
g : 0 < b.1 ∧ 0 < b.2 ∧ ∃ g, 0 < g ∧ ∃ x, 0 < x ∧ ∀ (n : ℕ), x ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = g
S : ∃ N, 0 < N ∧ ∀ (n : ℕ), N ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
D : 0 < Y ∧ 0 < L ∧ ∀ (n : ℕ), L ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
H : ¬b.1 * b.2 + 1 ∣ Y
w✝ : ℕ
h✝ : φ (b.1 * b.2 + 1) = 0 + w✝ + 1
this : (b.1 % (b.1 * b.2 + 1)) ^ w✝ * b.1 % (b.1 * b.2 + 1) = 1 ∧ (b.2 % (b.1 * b.2 + 1)) ^ w✝ * b.2 % (b.1 * b.2 + 1) = 1
⊢ ((1 % (b.1 * b.2 + 1)) ^ L * (b.1 % (b.1 * b.2 + 1)) ^ w✝ + b.2) % (b.1 * b.2 + 1) = 0 ∧
  ((1 % (b.1 * b.2 + 1)) ^ L * (b.2 % (b.1 * b.2 + 1)) ^ w✝ + b.1) % (b.1 * b.2 + 1) = 0
      /- Since a, b are coprime to ab + 1, it suffices to show that
        ab + 1 | a(a^(φ(ab + 1)(N + 1) - 1) + b) and
        ab + 1 | b(b^(φ(ab + 1)(N + 1) - 1) + a). -/
      suffices:b.1*b.2+1∣b.1*( (b.1%((b).1 * ( b.snd) + 1) : _)^‹Nat› +b.snd) ∧(b.fst * ↑(b.snd) + 1)∣(b).snd*( (b.snd%((b).fst * b.snd + 1))^ ‹Nat›+b.fst)this.intro
b : ℕ × ℕ
Y : ℕ
L : ℕ
g : 0 < b.1 ∧ 0 < b.2 ∧ ∃ g, 0 < g ∧ ∃ x, 0 < x ∧ ∀ (n : ℕ), x ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = g
S : ∃ N, 0 < N ∧ ∀ (n : ℕ), N ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
D : 0 < Y ∧ 0 < L ∧ ∀ (n : ℕ), L ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
H : ¬b.1 * b.2 + 1 ∣ Y
w✝ : ℕ
h✝ : φ (b.1 * b.2 + 1) = 0 + w✝ + 1
this✝ : (b.1 % (b.1 * b.2 + 1)) ^ w✝ * b.1 % (b.1 * b.2 + 1) = 1 ∧ (b.2 % (b.1 * b.2 + 1)) ^ w✝ * b.2 % (b.1 * b.2 + 1) = 1
this : b.1 * b.2 + 1 ∣ b.1 * ((b.1 % (b.1 * b.2 + 1)) ^ w✝ + b.2) ∧ b.1 * b.2 + 1 ∣ b.2 * ((b.2 % (b.1 * b.2 + 1)) ^ w✝ + b.1)
⊢ ((1 % (b.1 * b.2 + 1)) ^ L * (b.1 % (b.1 * b.2 + 1)) ^ w✝ + b.2) % (b.1 * b.2 + 1) = 0 ∧
  ((1 % (b.1 * b.2 + 1)) ^ L * (b.2 % (b.1 * b.2 + 1)) ^ w✝ + b.1) % (b.1 * b.2 + 1) = 0
this
b : ℕ × ℕ
Y : ℕ
L : ℕ
g : 0 < b.1 ∧ 0 < b.2 ∧ ∃ g, 0 < g ∧ ∃ x, 0 < x ∧ ∀ (n : ℕ), x ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = g
S : ∃ N, 0 < N ∧ ∀ (n : ℕ), N ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
D : 0 < Y ∧ 0 < L ∧ ∀ (n : ℕ), L ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
H : ¬b.1 * b.2 + 1 ∣ Y
w✝ : ℕ
h✝ : φ (b.1 * b.2 + 1) = 0 + w✝ + 1
this : (b.1 % (b.1 * b.2 + 1)) ^ w✝ * b.1 % (b.1 * b.2 + 1) = 1 ∧ (b.2 % (b.1 * b.2 + 1)) ^ w✝ * b.2 % (b.1 * b.2 + 1) = 1
⊢ b.1 * b.2 + 1 ∣ b.1 * ((b.1 % (b.1 * b.2 + 1)) ^ w✝ + b.2) ∧ b.1 * b.2 + 1 ∣ b.2 * ((b.2 % (b.1 * b.2 + 1)) ^ w✝ + b.1)
      ·this.intro
b : ℕ × ℕ
Y : ℕ
L : ℕ
g : 0 < b.1 ∧ 0 < b.2 ∧ ∃ g, 0 < g ∧ ∃ x, 0 < x ∧ ∀ (n : ℕ), x ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = g
S : ∃ N, 0 < N ∧ ∀ (n : ℕ), N ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
D : 0 < Y ∧ 0 < L ∧ ∀ (n : ℕ), L ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
H : ¬b.1 * b.2 + 1 ∣ Y
w✝ : ℕ
h✝ : φ (b.1 * b.2 + 1) = 0 + w✝ + 1
this✝ : (b.1 % (b.1 * b.2 + 1)) ^ w✝ * b.1 % (b.1 * b.2 + 1) = 1 ∧ (b.2 % (b.1 * b.2 + 1)) ^ w✝ * b.2 % (b.1 * b.2 + 1) = 1
this : b.1 * b.2 + 1 ∣ b.1 * ((b.1 % (b.1 * b.2 + 1)) ^ w✝ + b.2) ∧ b.1 * b.2 + 1 ∣ b.2 * ((b.2 % (b.1 * b.2 + 1)) ^ w✝ + b.1)
⊢ ((1 % (b.1 * b.2 + 1)) ^ L * (b.1 % (b.1 * b.2 + 1)) ^ w✝ + b.2) % (b.1 * b.2 + 1) = 0 ∧
  ((1 % (b.1 * b.2 + 1)) ^ L * (b.2 % (b.1 * b.2 + 1)) ^ w✝ + b.1) % (b.1 * b.2 + 1) = 0 norm_num[<-Nat.dvd_iff_mod_eq_zero,g,(1).mod_eq_of_lt,Nat.dvd_mul] at this⊢this.intro
b : ℕ × ℕ
Y : ℕ
L : ℕ
g : 0 < b.1 ∧ 0 < b.2 ∧ ∃ g, 0 < g ∧ ∃ x, 0 < x ∧ ∀ (n : ℕ), x ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = g
S : ∃ N, 0 < N ∧ ∀ (n : ℕ), N ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
D : 0 < Y ∧ 0 < L ∧ ∀ (n : ℕ), L ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
H : ¬b.1 * b.2 + 1 ∣ Y
w✝ : ℕ
h✝ : φ (b.1 * b.2 + 1) = 0 + w✝ + 1
this✝ : (b.1 % (b.1 * b.2 + 1)) ^ w✝ * b.1 % (b.1 * b.2 + 1) = 1 ∧ (b.2 % (b.1 * b.2 + 1)) ^ w✝ * b.2 % (b.1 * b.2 + 1) = 1
this : (∃ y, y ∣ b.1 ∧ ∃ x, x ∣ (b.1 % (b.1 * b.2 + 1)) ^ w✝ + b.2 ∧ y * x = b.1 * b.2 + 1) ∧
  ∃ y, y ∣ b.2 ∧ ∃ x, x ∣ (b.2 % (b.1 * b.2 + 1)) ^ w✝ + b.1 ∧ y * x = b.1 * b.2 + 1
⊢ b.1 * b.2 + 1 ∣ (b.1 % (b.1 * b.2 + 1)) ^ w✝ + b.2 ∧ b.1 * b.2 + 1 ∣ (b.2 % (b.1 * b.2 + 1)) ^ w✝ + b.1
        exists@?_this.intro.refine_1
b : ℕ × ℕ
Y : ℕ
L : ℕ
g : 0 < b.1 ∧ 0 < b.2 ∧ ∃ g, 0 < g ∧ ∃ x, 0 < x ∧ ∀ (n : ℕ), x ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = g
S : ∃ N, 0 < N ∧ ∀ (n : ℕ), N ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
D : 0 < Y ∧ 0 < L ∧ ∀ (n : ℕ), L ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
H : ¬b.1 * b.2 + 1 ∣ Y
w✝ : ℕ
h✝ : φ (b.1 * b.2 + 1) = 0 + w✝ + 1
this✝ : (b.1 % (b.1 * b.2 + 1)) ^ w✝ * b.1 % (b.1 * b.2 + 1) = 1 ∧ (b.2 % (b.1 * b.2 + 1)) ^ w✝ * b.2 % (b.1 * b.2 + 1) = 1
this : (∃ y, y ∣ b.1 ∧ ∃ x, x ∣ (b.1 % (b.1 * b.2 + 1)) ^ w✝ + b.2 ∧ y * x = b.1 * b.2 + 1) ∧
  ∃ y, y ∣ b.2 ∧ ∃ x, x ∣ (b.2 % (b.1 * b.2 + 1)) ^ w✝ + b.1 ∧ y * x = b.1 * b.2 + 1
⊢ b.1 * b.2 + 1 ∣ (b.1 % (b.1 * b.2 + 1)) ^ w✝ + b.2
this.intro.refine_2
b : ℕ × ℕ
Y : ℕ
L : ℕ
g : 0 < b.1 ∧ 0 < b.2 ∧ ∃ g, 0 < g ∧ ∃ x, 0 < x ∧ ∀ (n : ℕ), x ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = g
S : ∃ N, 0 < N ∧ ∀ (n : ℕ), N ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
D : 0 < Y ∧ 0 < L ∧ ∀ (n : ℕ), L ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
H : ¬b.1 * b.2 + 1 ∣ Y
w✝ : ℕ
h✝ : φ (b.1 * b.2 + 1) = 0 + w✝ + 1
this✝ : (b.1 % (b.1 * b.2 + 1)) ^ w✝ * b.1 % (b.1 * b.2 + 1) = 1 ∧ (b.2 % (b.1 * b.2 + 1)) ^ w✝ * b.2 % (b.1 * b.2 + 1) = 1
this : (∃ y, y ∣ b.1 ∧ ∃ x, x ∣ (b.1 % (b.1 * b.2 + 1)) ^ w✝ + b.2 ∧ y * x = b.1 * b.2 + 1) ∧
  ∃ y, y ∣ b.2 ∧ ∃ x, x ∣ (b.2 % (b.1 * b.2 + 1)) ^ w✝ + b.1 ∧ y * x = b.1 * b.2 + 1
⊢ b.1 * b.2 + 1 ∣ (b.2 % (b.1 * b.2 + 1)) ^ w✝ + b.1
        ·this.intro.refine_1
b : ℕ × ℕ
Y : ℕ
L : ℕ
g : 0 < b.1 ∧ 0 < b.2 ∧ ∃ g, 0 < g ∧ ∃ x, 0 < x ∧ ∀ (n : ℕ), x ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = g
S : ∃ N, 0 < N ∧ ∀ (n : ℕ), N ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
D : 0 < Y ∧ 0 < L ∧ ∀ (n : ℕ), L ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
H : ¬b.1 * b.2 + 1 ∣ Y
w✝ : ℕ
h✝ : φ (b.1 * b.2 + 1) = 0 + w✝ + 1
this✝ : (b.1 % (b.1 * b.2 + 1)) ^ w✝ * b.1 % (b.1 * b.2 + 1) = 1 ∧ (b.2 % (b.1 * b.2 + 1)) ^ w✝ * b.2 % (b.1 * b.2 + 1) = 1
this : (∃ y, y ∣ b.1 ∧ ∃ x, x ∣ (b.1 % (b.1 * b.2 + 1)) ^ w✝ + b.2 ∧ y * x = b.1 * b.2 + 1) ∧
  ∃ y, y ∣ b.2 ∧ ∃ x, x ∣ (b.2 % (b.1 * b.2 + 1)) ^ w✝ + b.1 ∧ y * x = b.1 * b.2 + 1
⊢ b.1 * b.2 + 1 ∣ (b.1 % (b.1 * b.2 + 1)) ^ w✝ + b.2 cases this.1 with|_ Q r=>this.intro.refine_1.intro
b : ℕ × ℕ
Y : ℕ
L : ℕ
g : 0 < b.1 ∧ 0 < b.2 ∧ ∃ g, 0 < g ∧ ∃ x, 0 < x ∧ ∀ (n : ℕ), x ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = g
S : ∃ N, 0 < N ∧ ∀ (n : ℕ), N ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
D : 0 < Y ∧ 0 < L ∧ ∀ (n : ℕ), L ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
H : ¬b.1 * b.2 + 1 ∣ Y
w✝ : ℕ
h✝ : φ (b.1 * b.2 + 1) = 0 + w✝ + 1
this✝ : (b.1 % (b.1 * b.2 + 1)) ^ w✝ * b.1 % (b.1 * b.2 + 1) = 1 ∧ (b.2 % (b.1 * b.2 + 1)) ^ w✝ * b.2 % (b.1 * b.2 + 1) = 1
this : (∃ y, y ∣ b.1 ∧ ∃ x, x ∣ (b.1 % (b.1 * b.2 + 1)) ^ w✝ + b.2 ∧ y * x = b.1 * b.2 + 1) ∧
  ∃ y, y ∣ b.2 ∧ ∃ x, x ∣ (b.2 % (b.1 * b.2 + 1)) ^ w✝ + b.1 ∧ y * x = b.1 * b.2 + 1
Q : ℕ
r : Q ∣ b.1 ∧ ∃ x, x ∣ (b.1 % (b.1 * b.2 + 1)) ^ w✝ + b.2 ∧ Q * x = b.1 * b.2 + 1
⊢ b.1 * b.2 + 1 ∣ (b.1 % (b.1 * b.2 + 1)) ^ w✝ + b.2simp_all[(Q.dvd_gcd r.1 ⟨_,.symm r.right.choose_spec.2⟩).antisymm]All goals completed! 🐙
        cases@this.2with|_ F X=>this.intro.refine_2.intro
b : ℕ × ℕ
Y : ℕ
L : ℕ
g : 0 < b.1 ∧ 0 < b.2 ∧ ∃ g, 0 < g ∧ ∃ x, 0 < x ∧ ∀ (n : ℕ), x ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = g
S : ∃ N, 0 < N ∧ ∀ (n : ℕ), N ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
D : 0 < Y ∧ 0 < L ∧ ∀ (n : ℕ), L ≤ n → (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
H : ¬b.1 * b.2 + 1 ∣ Y
w✝ : ℕ
h✝ : φ (b.1 * b.2 + 1) = 0 + w✝ + 1
this✝ : (b.1 % (b.1 * b.2 + 1)) ^ w✝ * b.1 % (b.1 * b.2 + 1) = 1 ∧ (b.2 % (b.1 * b.2 + 1)) ^ w✝ * b.2 % (b.1 * b.2 + 1) = 1
this : (∃ y, y ∣ b.1 ∧ ∃ x, x ∣ (b.1 % (b.1 * b.2 + 1)) ^ w✝ + b.2 ∧ y * x = b.1 * b.2 + 1) ∧
  ∃ y, y ∣ b.2 ∧ ∃ x, x ∣ (b.2 % (b.1 * b.2 + 1)) ^ w✝ + b.1 ∧ y * x = b.1 * b.2 + 1
F : ℕ
X : F ∣ b.2 ∧ ∃ x, x ∣ (b.2 % (b.1 * b.2 + 1)) ^ w✝ + b.1 ∧ F * x = b.1 * b.2 + 1
⊢ b.1 * b.2 + 1 ∣ (b.2 % (b.1 * b.2 + 1)) ^ w✝ + b.1simp_all[(F.dvd_gcd X.1 ⟨_,symm X.2.choose_spec.2⟩).antisymm]All goals completed! 🐙
      /- This follows from Euler's Theorem:
        a(a^(φ(ab + 1)(N + 1) - 1) + b)
        ≡ a^(φ(ab + 1)(N + 1)) + ab
        ≡ 1 + ab
        ≡ 0 (mod ab + 1)
        and similarly for b(b^(φ(ab + 1)(N + 1) - 1) + a). -/
      simp_all[mul_comm, mul_add,add_comm,Nat.add_mod,Nat.dvd_iff_mod_eq_zero]All goals completed! 🐙
    repeat use(Nat.ModEq.pow_totient (byb : ℕ × ℕ
g : b ∈ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g}
Y : ℕ
S : ∃ N, 0 < Y ∧ 0 < N ∧ ∀ n ≥ N, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
L : ℕ
D : 0 < Y ∧ 0 < L ∧ ∀ n ≥ L, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
H : ¬b.1 * b.2 + 1 ∣ Y
⊢ b.2.Coprime (b.1 * b.2 + 1) .b : ℕ × ℕ
g : b ∈ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g}
Y : ℕ
S : ∃ N, 0 < Y ∧ 0 < N ∧ ∀ n ≥ N, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
L : ℕ
D : 0 < Y ∧ 0 < L ∧ ∀ n ≥ L, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
H : ¬b.1 * b.2 + 1 ∣ Y
⊢ b.2.Coprime (b.1 * b.2 + 1) .b : ℕ × ℕ
g : b ∈ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g}
Y : ℕ
S : ∃ N, 0 < Y ∧ 0 < N ∧ ∀ n ≥ N, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
L : ℕ
D : 0 < Y ∧ 0 < L ∧ ∀ n ≥ L, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
H : ¬b.1 * b.2 + 1 ∣ Y
⊢ b.2.Coprime (b.1 * b.2 + 1) .b : ℕ × ℕ
g : b ∈ {(a, b) | 0 < a ∧ 0 < b ∧ ∃ g N, 0 < g ∧ 0 < N ∧ ∀ n ≥ N, (a ^ n + b).gcd (b ^ n + a) = g}
Y : ℕ
S : ∃ N, 0 < Y ∧ 0 < N ∧ ∀ n ≥ N, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
i : ¬b = (1, 1)
L : ℕ
D : 0 < Y ∧ 0 < L ∧ ∀ n ≥ L, (b.1 ^ n + b.2).gcd (b.2 ^ n + b.1) = Y
H : ¬b.1 * b.2 + 1 ∣ Y
⊢ b.2.Coprime (b.1 * b.2 + 1)norm_numAll goals completed! 🐙) )
  congr 26All goals completed! 🐙

The following command shows which axioms the proof relies upon:

'imo_2024_p2' depends on axioms: [propext, Classical.choice, Quot.sound]#print axioms imo_2024_p2

'imo_2024_p2' depends on axioms: [propext, Classical.choice, Quot.sound]

These are the standard built in axioms.