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

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

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-oeis-308734,
  title        = {Four-square conjecture with powers of 2, 3, and 5},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-308734}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

Also: CITATION.cff · Atom feed of results

Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

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 $a, b, c, d, x, y$ .

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.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«308734».

theorem conjecture (n : ℕ) (hn : 1 < n) : A n

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

A308734
Z.-W. Sun, "Refining Lagrange's four-square theorem," J. Number Theory 175 (2017), 167-190. https://doi.org/10.1016/j.jnt.2016.11.008
Z.-W. Sun, "Restricted sums of four squares," Int. J. Number Theory 15 (2019), 1863-1893.
Z.-W. Sun, "Various Refinements of Lagrange's Four-Square Theorem," Westlake Number Theory Symposium, Nanjing University, China, 2020.
S. Banerjee, "On a conjecture of Sun about sums of restricted squares," J. Number Theory 256 (2024), 253-289.

Source and licence

Imported from Formal Conjectures (OEIS), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.