Sum of a triangular number, a generalized pentagonal number, and a generalized heptagonal number

formal-conjectures · oeis · ams-11

Any nonnegative integer can be written as $x(x+1)/2 + y(3y+1)/2 + z(5z+1)/2$ with $x, y, z$
nonnegative integers.

Zhi-Wei Sun has offered a USD 135 prize for the first proof of this conjecture.

**Zhi-Wei Sun's Conjecture (A287616)**: Any nonnegative integer can be written as the sum of
a triangular number $x(x+1)/2$, a generalized pentagonal number $y(3y+1)/2$, and a generalized
heptagonal number $z(5z+1)/2$, where $x, y, z$ are nonnegative integers.

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/A287616
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/OEIS/287616.lean
Local target: Corpus.OEIS287616.conjecture
Source SHA-256: 2c9ea9d3ede7d18c225dcb121d387348db787692dab7137cce66f099ff4709d1
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