Fermat's Last Theorem for the case n = 4

There are no non-zero integers a, b and c such that a ^ 4 + b ^ 4 = c ^ 4.

def Fermat42 (a : ℤ) (b : ℤ) (c : ℤ) : Prop

Shorthand for three non-zero integers a, b, and c satisfying a ^ 4 + b ^ 4 = c ^ 2. We will show that no integers satisfy this equation. Clearly Fermat's Last theorem for n = 4 follows.

Equations

• Fermat42 a b c = (a ≠ 0 ∧ b ≠ 0 ∧ a ^ 4 + b ^ 4 = c ^ 2)

theorem Fermat42.comm {a : ℤ} {b : ℤ} {c : ℤ} : Fermat42 a b c ↔ Fermat42 b a c

theorem Fermat42.mul {a : ℤ} {b : ℤ} {c : ℤ} {k : ℤ} (hk0 : k ≠ 0) : Fermat42 a b c ↔ Fermat42 (k * a) (k * b) (k ^ 2 * c)

theorem Fermat42.ne_zero {a : ℤ} {b : ℤ} {c : ℤ} (h : Fermat42 a b c) : c ≠ 0

def Fermat42.Minimal (a : ℤ) (b : ℤ) (c : ℤ) : Prop

We say a solution to a ^ 4 + b ^ 4 = c ^ 2 is minimal if there is no other solution with a smaller c (in absolute value).

Equations

• Fermat42.Minimal a b c = (Fermat42 a b c ∧ ∀ (a1 b1 c1 : ℤ), Fermat42 a1 b1 c1 → c.natAbs ≤ c1.natAbs)

theorem Fermat42.exists_minimal {a : ℤ} {b : ℤ} {c : ℤ} (h : Fermat42 a b c) : ∃ (a0 : ℤ) (b0 : ℤ) (c0 : ℤ), Fermat42.Minimal a0 b0 c0

if we have a solution to a ^ 4 + b ^ 4 = c ^ 2 then there must be a minimal one.

theorem Fermat42.coprime_of_minimal {a : ℤ} {b : ℤ} {c : ℤ} (h : Fermat42.Minimal a b c) : IsCoprime a b

a minimal solution to a ^ 4 + b ^ 4 = c ^ 2 must have a and b coprime.

theorem Fermat42.minimal_comm {a : ℤ} {b : ℤ} {c : ℤ} : Fermat42.Minimal a b c → Fermat42.Minimal b a c

We can swap a and b in a minimal solution to a ^ 4 + b ^ 4 = c ^ 2.

theorem Fermat42.neg_of_minimal {a : ℤ} {b : ℤ} {c : ℤ} : Fermat42.Minimal a b c → Fermat42.Minimal a b (-c)

We can assume that a minimal solution to a ^ 4 + b ^ 4 = c ^ 2 has positive c.

theorem Fermat42.exists_odd_minimal {a : ℤ} {b : ℤ} {c : ℤ} (h : Fermat42 a b c) : ∃ (a0 : ℤ) (b0 : ℤ) (c0 : ℤ), Fermat42.Minimal a0 b0 c0 ∧ a0 % 2 = 1

We can assume that a minimal solution to a ^ 4 + b ^ 4 = c ^ 2 has a odd.

theorem Fermat42.exists_pos_odd_minimal {a : ℤ} {b : ℤ} {c : ℤ} (h : Fermat42 a b c) : ∃ (a0 : ℤ) (b0 : ℤ) (c0 : ℤ), Fermat42.Minimal a0 b0 c0 ∧ a0 % 2 = 1 ∧ 0 < c0

We can assume that a minimal solution to a ^ 4 + b ^ 4 = c ^ 2 has a odd and c positive.

theorem Int.coprime_of_sq_sum {r : ℤ} {s : ℤ} (h2 : IsCoprime s r) : IsCoprime (r ^ 2 + s ^ 2) r

theorem Int.coprime_of_sq_sum' {r : ℤ} {s : ℤ} (h : IsCoprime r s) : IsCoprime (r ^ 2 + s ^ 2) (r * s)

theorem Fermat42.not_minimal {a : ℤ} {b : ℤ} {c : ℤ} (h : Fermat42.Minimal a b c) (ha2 : a % 2 = 1) (hc : 0 < c) : False

theorem not_fermat_42 {a : ℤ} {b : ℤ} {c : ℤ} (ha : a ≠ 0) (hb : b ≠ 0) : a ^ 4 + b ^ 4 ≠ c ^ 2

theorem fermatLastTheoremFour : FermatLastTheoremFor 4

Fermat's Last Theorem for $n=4$: if a b c : ℕ are all non-zero then a ^ 4 + b ^ 4 ≠ c ^ 4.

theorem FermatLastTheorem.of_odd_primes (hprimes : ∀ (p : ℕ), p.Prime → Odd p → FermatLastTheoremFor p) : FermatLastTheorem

To prove Fermat's Last Theorem, it suffices to prove it for odd prime exponents.