Representations with prime conditions

Zhi-Wei Sun's Conjecture (A232174): Any integer n > 1 can be written as x + y with x, y > 0 such that both x + ny and x^2 + ny^2 are prime.

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-232174,
  title        = {Representations with prime conditions},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-232174}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Zhi-Wei Sun's Conjecture (A232174): Any integer $n > 1$ can be written as $x + y$ with $x, y > 0$ such that both $x + n y$ and $x^{2} + n y^{2}$ are prime.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«232174».

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

A232174
Z.-W. Sun, "Conjectures on representations involving primes," in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II: CANT, Springer Proc. in Math. & Stat., Vol. 220, Springer, 2017, pp. 279-310. https://arxiv.org/abs/1211.1588
D.A. Cox, "Primes of the Form x² + ny²," John Wiley & Sons, 1989.

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.