1. FormalConjectures
  2. OEIS
  3. «281976»
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Sum of four squares with square conditions

Any integer $n \geq 0$ can be written as $x^2 + y^2 + z^2 + w^2$ with $x, y, z, w$ nonnegative integers and $z \leq w$, such that both $x$ and $x + 24y$ are squares.

Zhi-Wei Sun has offered a $2,400 prize for the first proof.

References:

    A281976

    Z.-W. Sun, "Refining Lagrange's four-square theorem," J. Number Theory 175 (2017), 167-190. https://doi.org/10.1016/j.jnt.2016.11.008

    Z.-W. Sun, "Restricted sums of four squares," arXiv:1701.05868 [math.NT], 2017. https://arxiv.org/abs/1701.05868

namespace OeisA281976

The predicate that n can be written as $x^2 + y^2 + z^2 + w^2$ with $x, y, z, w$ nonnegative integers, $z \leq w$, such that both $x$ and $x + 24y$ are squares.

def A (n : ℕ) : Prop := ∃ x y z w : ℕ, n = x ^ 2 + y ^ 2 + z ^ 2 + w ^ 2 ∧ z ≤ w ∧ IsSquare x ∧ IsSquare (x + 24 * y) @[category test, AMS 11] theorem a_0 : A 0 := ⟨0, 0, 0, 0, ⊢ 0 = 0 ^ 2 + 0 ^ 2 + 0 ^ 2 + 0 ^ 2 ∧ 0 ≤ 0 ∧ IsSquare 0 ∧ IsSquare (0 + 24 * 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! 🐙, ⊢ 0 ≤ 0 All goals completed! 🐙, ⟨1, ⊢ 1 = 1 * 1 All goals completed! 🐙⟩, ⟨1, ⊢ 1 + 24 * 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! 🐙, ⊢ 0 ≤ 1 All goals completed! 🐙, ⟨1, ⊢ 1 = 1 * 1 All goals completed! 🐙⟩, ⟨1, ⊢ 1 + 24 * 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 All goals completed! 🐙, ⟨1, ⊢ 1 = 1 * 1 All goals completed! 🐙⟩, ⟨1, ⊢ 1 + 24 * 0 = 1 * 1 All goals completed! 🐙⟩⟩ @[category test, AMS 11] theorem a_4 : A 4 := ⟨0, 0, 0, 2, ⊢ 4 = 0 ^ 2 + 0 ^ 2 + 0 ^ 2 + 2 ^ 2 All goals completed! 🐙, ⊢ 0 ≤ 2 All goals completed! 🐙, ⟨0, ⊢ 0 = 0 * 0 All goals completed! 🐙⟩, ⟨0, ⊢ 0 + 24 * 0 = 0 * 0 All goals completed! 🐙⟩⟩ @[category test, AMS 11] theorem a_8 : A 8 := ⟨0, 0, 2, 2, ⊢ 8 = 0 ^ 2 + 0 ^ 2 + 2 ^ 2 + 2 ^ 2 All goals completed! 🐙, ⊢ 2 ≤ 2 All goals completed! 🐙, ⟨0, ⊢ 0 = 0 * 0 All goals completed! 🐙⟩, ⟨0, ⊢ 0 + 24 * 0 = 0 * 0 All goals completed! 🐙⟩⟩ @[category test, AMS 11] theorem a_12 : A 12 := ⟨1, 1, 1, 3, ⊢ 12 = 1 ^ 2 + 1 ^ 2 + 1 ^ 2 + 3 ^ 2 All goals completed! 🐙, ⊢ 1 ≤ 3 All goals completed! 🐙, ⟨1, ⊢ 1 = 1 * 1 All goals completed! 🐙⟩, ⟨5, ⊢ 1 + 24 * 1 = 5 * 5 All goals completed! 🐙⟩⟩ @[category test, AMS 11] theorem a_23 : A 23 := ⟨1, 2, 3, 3, ⊢ 23 = 1 ^ 2 + 2 ^ 2 + 3 ^ 2 + 3 ^ 2 All goals completed! 🐙, ⊢ 3 ≤ 3 All goals completed! 🐙, ⟨1, ⊢ 1 = 1 * 1 All goals completed! 🐙⟩, ⟨7, ⊢ 1 + 24 * 2 = 7 * 7 All goals completed! 🐙⟩⟩ @[category test, AMS 11] theorem a_24 : A 24 := ⟨4, 0, 2, 2, ⊢ 24 = 4 ^ 2 + 0 ^ 2 + 2 ^ 2 + 2 ^ 2 All goals completed! 🐙, ⊢ 2 ≤ 2 All goals completed! 🐙, ⟨2, ⊢ 4 = 2 * 2 All goals completed! 🐙⟩, ⟨2, ⊢ 4 + 24 * 0 = 2 * 2 All goals completed! 🐙⟩⟩

Zhi-Wei Sun's Conjecture (A281976): Any integer $n \geq 0$ can be written as $x^2 + y^2 + z^2 + w^2$ with $x, y, z, w$ nonnegative integers and $z \leq w$, such that both $x$ and $x + 24y$ are squares.

@[category research open, AMS 11] theorem declaration uses 'sorry'conjecture (n : ℕ) : A n := n:ℕ⊢ A n All goals completed! 🐙 end OeisA281976