Lagrange's four square theorem

The main result in this file is sum_four_squares, a proof that every natural number is the sum of four square numbers.

Implementation Notes

The proof used is close to Lagrange's original proof.

theorem euler_four_squares {R : Type u_1} [CommRing R] (a : R) (b : R) (c : R) (d : R) (x : R) (y : R) (z : R) (w : R) : (a * x - b * y - c * z - d * w) ^ 2 + (a * y + b * x + c * w - d * z) ^ 2 + (a * z - b * w + c * x + d * y) ^ 2 + (a * w + b * z - c * y + d * x) ^ 2 = (a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2) * (x ^ 2 + y ^ 2 + z ^ 2 + w ^ 2)

Euler's four-square identity.

theorem Nat.euler_four_squares (a : ℕ) (b : ℕ) (c : ℕ) (d : ℕ) (x : ℕ) (y : ℕ) (z : ℕ) (w : ℕ) : Int.natAbs (↑a * ↑x - ↑b * ↑y - ↑c * ↑z - ↑d * ↑w) ^ 2 + Int.natAbs (↑a * ↑y + ↑b * ↑x + ↑c * ↑w - ↑d * ↑z) ^ 2 + Int.natAbs (↑a * ↑z - ↑b * ↑w + ↑c * ↑x + ↑d * ↑y) ^ 2 + Int.natAbs (↑a * ↑w + ↑b * ↑z - ↑c * ↑y + ↑d * ↑x) ^ 2 = (a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2) * (x ^ 2 + y ^ 2 + z ^ 2 + w ^ 2)

Euler's four-square identity, a version for natural numbers.

theorem Int.sq_add_sq_of_two_mul_sq_add_sq {m : ℤ} {x : ℤ} {y : ℤ} (h : 2 * m = x ^ 2 + y ^ 2) : m = ((x - y) / 2) ^ 2 + ((x + y) / 2) ^ 2

theorem Int.lt_of_sum_four_squares_eq_mul {a : ℕ} {b : ℕ} {c : ℕ} {d : ℕ} {k : ℕ} {m : ℕ} (h : a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = k * m) (ha : 2 * a < m) (hb : 2 * b < m) (hc : 2 * c < m) (hd : 2 * d < m) : k < m

theorem Int.exists_sq_add_sq_add_one_eq_mul (p : ℕ) [hp : Fact (Nat.Prime p)] : ∃ (a : ℕ) (b : ℕ) (k : ℕ), 0 < k ∧ k < p ∧ a ^ 2 + b ^ 2 + 1 = k * p

@[deprecated Int.exists_sq_add_sq_add_one_eq_mul]

theorem Int.exists_sq_add_sq_add_one_eq_k (p : ℕ) [Fact (Nat.Prime p)] : ∃ (a : ℤ) (b : ℤ) (k : ℕ), a ^ 2 + b ^ 2 + 1 = ↑k * ↑p ∧ k < p

theorem Nat.Prime.sum_four_squares {p : ℕ} (hp : Nat.Prime p) : ∃ (a : ℕ) (b : ℕ) (c : ℕ) (d : ℕ), a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = p

Lagrange's four squares theorem for a prime number. Use Nat.sum_four_squares instead.

theorem Nat.sum_four_squares (n : ℕ) : ∃ (a : ℕ) (b : ℕ) (c : ℕ) (d : ℕ), a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = n

Four squares theorem