The 2-4-6-8 Conjecture

formal-conjectures · oeis · ams-11

Any integer $n > 0$ can be written as $\binom{w+2}{2} + \binom{x+3}{4} + \binom{y+5}{6} + \binom{z+7}{8}$
with $w, x, y, z$ nonnegative integers.

Zhi-Wei Sun has offered a $2,468 prize for the first proof (or $2,468 RMB for a counterexample).

The conjecture has been verified for all $n$ up to $1.2 \times 10^{12}$ by Yaakov Baruch (March 2019).

**Zhi-Wei Sun's 2-4-6-8 Conjecture (A306477)**: Any integer $n > 0$ can be written as
$\binom{w+2}{2} + \binom{x+3}{4} + \binom{y+5}{6} + \binom{z+7}{8}$ for nonnegative integers $w, x, y, z$.

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/A306477
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://mathoverflow.net/questions/323541
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/OEIS/306477.lean
Local target: Corpus.OEIS306477.conjecture
Source SHA-256: 762b09119ebe6116026abe9e59ab7a74ed7656fb81ade44c1dd17b81f498dfdc
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