Sum of four squares with square conditions

formal-conjectures · oeis · ams-11

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.

**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.

Mathematical status
Open: marked research open in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11).

Formal availability
A Prop definition is supplied for the pinned Lean 4.33.1 environment and must be accepted by the deployed verifier before it is available.

Sources and provenance
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://oeis.org/A281976
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://doi.org/10.1016/j.jnt.2016.11.008
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://arxiv.org/abs/1701.05868
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/OEIS/281976.lean
Local target: Corpus.OEIS281976.conjecture
Source SHA-256: e2b53fe2d140f2030ed448c31061ef40573ecf64d09db89fadb4f60a1608c100
Lean v4.33.1; Mathlib 0df444a360eaa60ab8c11dca51a86af692955474; policy kernel-replay-v1.
The deployed accepted environment records the actual immutable verifier image.

Research discussions are not verified proofs. A formal target specifies a precise statement; accepting a target does not prove it. Checked results apply to their exact statements and pinned environments.

Public JSON record