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Claim. The preprint Entropy-Sieve Methods and Energy Functionals in the Erdős Problem [Er79] on Quadratic Prime Representations claims that only finitely many integers are not of the form $ap^2+b$ with $p$ prime, $a\ge1$ and $0\le b<p$, so that Problem 676 would have a positive answer. The claim assumes the paper's Strong Uniformity Hypothesis. This unproved statement says that the residues of $n$ modulo the squares of small primes are distributed almost independently, as measured by a quadratic energy and a relative entropy. The paper asserts that the hypothesis follows from the Elliott-Halberstam conjecture or the generalized Riemann hypothesis. The hypothesis and that derivation are both part of the claim.

Dispute. A thread post of 2025-12-05 reports that an AI reading (ChatGPT Pro) flagged potential issues, and it advises waiting until a journal accepts the paper. A post of 2025-12-26 reports a ChatGPT reading with three objections. First, the finiteness step, a Borel-Cantelli argument in the paper's Theorem 6.9, gives at most an average bound. Second, near-independence of the residues predicts about $x/\log x$ exceptions up to $x$, not finitely many. Third, the Elliott-Halberstam conjecture and GRH concern primes in progressions and do not bear on the hypothesis as the paper defines it. The claimant's later postings predict infinitely many exceptions. A thread post of 2026-05-24 gives a heuristic count of order $x/\log x$, and the 2026 write-up conjectures $x^{2/3+o(1)}$ exceptions in the version with any modulus. No acceptance evidence is on record: there is no journal publication, review or formalization, and the site labels the problem OPEN.

Depends on. No other wiki page; the claim rests on the preprint and its unproved hypothesis.