1. FormalConjectures
  2. OEIS
  3. «280831»
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The 1680-Conjecture

Any nonnegative integer can be written as $x^2 + y^2 + z^2 + w^2$ with $x, y, z, w$ nonnegative integers such that $x^4 + 1680 y^3 z$ is a square.

Zhi-Wei Sun has offered a prize of 1,680 RMB for the first proof.

References:

    A280831

    Z.-W. Sun, "Refining Lagrange's four-square theorem," J. Number Theory 175 (2017), 167-190.

    Z.-W. Sun, "Refining Lagrange's four-square theorem," arXiv:1604.06723 [math.NT], 2016.

namespace OeisA280831

The predicate that n can be written as $x^2 + y^2 + z^2 + w^2$ with $x, y, z, w$ nonnegative integers such that $x^4 + 1680 y^3 z$ is a square.

def A (n : ℕ) : Prop := ∃ x y z w : ℕ, n = x ^ 2 + y ^ 2 + z ^ 2 + w ^ 2 ∧ IsSquare (x ^ 4 + 1680 * y ^ 3 * z) @[category test, AMS 11] theorem a_0 : A 0 := ⟨0, 0, 0, 0, ⊢ 0 = 0 ^ 2 + 0 ^ 2 + 0 ^ 2 + 0 ^ 2 All goals completed! 🐙, 0, ⊢ 0 ^ 4 + 1680 * 0 ^ 3 * 0 = 0 * 0 All goals completed! 🐙⟩ @[category test, AMS 11] theorem a_1 : A 1 := ⟨1, 0, 0, 0, ⊢ 1 = 1 ^ 2 + 0 ^ 2 + 0 ^ 2 + 0 ^ 2 All goals completed! 🐙, 1, ⊢ 1 ^ 4 + 1680 * 0 ^ 3 * 0 = 1 * 1 All goals completed! 🐙⟩ @[category test, AMS 11] theorem a_2 : A 2 := ⟨1, 0, 0, 1, ⊢ 2 = 1 ^ 2 + 0 ^ 2 + 0 ^ 2 + 1 ^ 2 All goals completed! 🐙, 1, ⊢ 1 ^ 4 + 1680 * 0 ^ 3 * 0 = 1 * 1 All goals completed! 🐙⟩ @[category test, AMS 11] theorem a_3 : A 3 := ⟨1, 0, 1, 1, ⊢ 3 = 1 ^ 2 + 0 ^ 2 + 1 ^ 2 + 1 ^ 2 All goals completed! 🐙, 1, ⊢ 1 ^ 4 + 1680 * 0 ^ 3 * 1 = 1 * 1 All goals completed! 🐙⟩ @[category test, AMS 11] theorem a_4 : A 4 := ⟨2, 0, 0, 0, ⊢ 4 = 2 ^ 2 + 0 ^ 2 + 0 ^ 2 + 0 ^ 2 All goals completed! 🐙, 4, ⊢ 2 ^ 4 + 1680 * 0 ^ 3 * 0 = 4 * 4 All goals completed! 🐙⟩ @[category test, AMS 11] theorem a_7 : A 7 := ⟨1, 1, 1, 2, ⊢ 7 = 1 ^ 2 + 1 ^ 2 + 1 ^ 2 + 2 ^ 2 All goals completed! 🐙, 41, ⊢ 1 ^ 4 + 1680 * 1 ^ 3 * 1 = 41 * 41 All goals completed! 🐙⟩ @[category test, AMS 11] theorem a_95 : A 95 := ⟨6, 3, 1, 7, ⊢ 95 = 6 ^ 2 + 3 ^ 2 + 1 ^ 2 + 7 ^ 2 All goals completed! 🐙, 216, ⊢ 6 ^ 4 + 1680 * 3 ^ 3 * 1 = 216 * 216 All goals completed! 🐙⟩

Zhi-Wei Sun's 1680-Conjecture (A280831): Any nonnegative integer can be written as $x^2 + y^2 + z^2 + w^2$ with $x, y, z, w$ nonnegative integers such that $x^4 + 1680 y^3 z$ is a square.

@[category research open, AMS 11] theorem declaration uses 'sorry'conjecture (n : ℕ) : A n := n:ℕ⊢ A n All goals completed! 🐙 end OeisA280831