Four-square conjecture with powers of 2, 3, and 5

formal-conjectures · oeis · ams-11

Any integer $n > 1$ can be written as $(2^a \cdot 3^b)^2 + (2^c \cdot 5^d)^2 + x^2 + y^2$
where $a, b, c, d, x, y$ are nonnegative integers.

Zhi-Wei Sun has offered a \$2,500 prize for the first proof.

**Zhi-Wei Sun's Four-Square Conjecture (A308734)**: Any integer $n > 1$ can be written as
$(2^a \cdot 3^b)^2 + (2^c \cdot 5^d)^2 + x^2 + y^2$ for nonnegative integers $a, b, c, d, x, y$.

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/A308734
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://doi.org/10.1016/j.jnt.2016.11.008
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/OEIS/308734.lean
Local target: Corpus.OEIS308734.conjecture
Source SHA-256: 5b153489b485b249abf608af1eb26fff90b12c4bfb56f040f48bc008e0f43897
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