Problem 279

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claims/: The 1 claim page of Problem 279, one per claimant's result; the problem's standing derives from them.

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Statement. Let $k\geq 3$. Is there a choice of congruence classes $a_p\pmod{p}$ for every prime $p$ such that all sufficiently large integers can be written as $a_p+tp$ for some prime $p$ and integer $t\geq k$?

Formulation. The site, like its source (Erdős and Graham's 1980 monograph, p. 29), does not say which integer represents each class, and the condition $t\ge k$ needs one: with $a_p=-kp$, every $n\ge2$ is $a_p+tp$ for a prime factor $p$ of $n$ and $t=k+n/p$. The page reads $a_p$ as the least residue, $0\le a_p<p$, the reading under which the source's remark that the condition $t\ge1$ already rules out some sequences of moduli makes sense, and the normalization Chen's manuscript states.

Status. Open. The site labels the problem OPEN (page last edited 17 April 2026). Wanfang Chen claims a yes answer for every $k\ge3$, under the least-residue reading, in a manuscript posted with a Lean 4 development on 2026-07-28; the manuscript says the proof was generated by OpenAI's GPT-5.6-sol model. The work was pointed to on the site's discussion thread on 2026-09-05, the curator has not ruled on it, and it is recorded on Chen's claim page. Przemek Chojecki's note of 16 April 2026, linked on the thread, proves a density-one weakening (a gap in its Theorem 7 was pointed out on the thread) and conditional implications between cases; it settles no case of the question, and its author asked that it be disregarded once the site corrected its remark, so it has no claim page.

Source. erdosproblems.com/279, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #279, https://www.erdosproblems.com/279.

Formalization. Statement in formal-conjectures.

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