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Claim. The posting is filed on the site's proof-claims tab for Problem 676 as a full proof claim, submitted 2026-07-25 by Rafik Zeraoulia, who names the AI system OpenAI GPT-5.6 Thinking as having been used. The claimant asserts neither answer to the question: the claim's own summary on the tab describes the work as progress that stops short of proving the conjecture, and the abstract of the write-up, Barrier reformulations and computational progress on an Erdős representation problem (Zenodo record 21560330, version 1, published 2026-07-25), states that whether the representation holds for every sufficiently large integer remains open. The page is therefore recorded as withdrawn below a proof, and the problem's standing takes nothing from it. Its claim value records the direction of the work's evidence, a negative answer: the computed exceptions are exceptions to the question's form, and the write-up conjectures that their count grows like a power of $x$.

Submission note. Posted to erdosproblems.com as a proof claim by Rafik Zeraoulia (account Rafikzeraoulia2025) on 25 July 2026, giving "OpenAI GPT-5.6 Thinking" as the AI used:

This is partial progress rather than a proof of the original conjecture. The representation condition is rewritten as [ n \bmod m^2 < m, ] equivalently, [ m \mid \left\lfloor \frac{n}{m} \right\rfloor. ] This leads to a barrier formulation involving the largest square divisor of an integer. The paper proves a density criterion for pairwise coprime moduli, establishes a positive-correlation inequality for the relevant congruence events, and gives an exact (O(X)) interval-sieve algorithm for finding exceptions. Computations up to (10^9), together with searches in selected larger intervals, produce many verified exceptions in the unrestricted-modulus version, including [ 10000005783830. ] Notes: The original prime problem remains open. The paper does not claim that infinitely many exceptions exist. It reports rigorous reformulations, partial density results, reproducible computations, and the numerical conjecture [ E_{\mathrm{all}}(x)=x^{2/3+o(1)} ] for the counting function of exceptions in the unrestricted-modulus problem.

What the write-up asserts. As the summary and abstract describe it, the write-up rewrites the condition that $n$ be of the form $am^2+b$ with $0\le b<m$ as $n\bmod m^2<m$, equivalently $m\mid\lfloor n/m\rfloor$, and restates it through a barrier function involving the largest square divisor of an integer; it proves a density criterion for pairwise coprime moduli and a positive-correlation inequality for the congruence events involved; it gives a linear-time interval sieve for exceptions and reports exhaustive computations to $10^9$ together with searches in selected larger intervals, producing many exceptions in the variant where the modulus $m$ ranges over all integers at least $2$ rather than over primes, among them $n=10000005783830$ and $150$ exceptions in $[10^{13},10^{13}+2\cdot10^7)$, with a conjectured growth law $x^{2/3+o(1)}$ for their count. None of these statements settles any part of the question, which asks about all sufficiently large integers; a finite list of exceptions does not decide it, and the growth law is a conjecture. The claimant asserts no reduction of the problem to another statement, so no partial page is recorded.

Exceptions and acceptance. For $n=10000005783830$, no prime $p$ with $p^2\le n$ satisfies $n\bmod p^2<p$, and no integer $m\ge2$ does either, so $n$ is not of the form $ap^2+b$ with $a\ge1$ and $0\le b<p$ for any prime $p$ (a prime $p>\sqrt n$ would force $a=0$). Since an exception for every modulus $m\ge2$ is in particular an exception for every prime, the unrestricted-modulus exceptions are exceptions to the question's form. No acceptance evidence of any kind is on record: the site's label is OPEN (page last edited 2026-04-07), the tab's claim had no comments on 2026-10-06, and there is no refereed publication, independent review or formalization.

Depends on. No other wiki page; the posting rests on the write-up above.