The 1680-Conjecture

formal-conjectures · oeis · ams-11

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.

**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.

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/A280831
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/OEIS/280831.lean
Local target: Corpus.OEIS280831.conjecture
Source SHA-256: 37806943ac8fda54257ed8dc5f740c67e72a1ab11c1569e56b0ff50178a81f41
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