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Claim. P. G. Walsh, A question of Erdős on $3$-powerful numbers and an elliptic curve analogue of the Ankeny-Artin-Chowla conjecture, Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 29 (2024), 83--87, arXiv:2404.03970, cited as [Wa24] on the problem page. Its Theorem 1.1: if $p$ is an odd prime for which the curve $E:Y^2=X^3-432p^2$ has positive rank, then $x^3+y^3=p^4z^3$ has infinitely many pairwise coprime integer solutions, which can be derived from the multiples $(3pk)P$, $k\geq1$, of a generator $P$ of infinite order. Each such solution gives, after the negative terms are moved across, three pairwise coprime $3$-powerful numbers $a,b,c$ with $a+b=c$, the third question of Problem 939. The statement is read in the arXiv version (v1, 5 April 2024).

The hypothesis is unproved in the paper: it exhibits no prime $p$ at which $E$ has positive rank, and records Adam Logan's remark that the $3$-Selmer group of $E$ is trivial for $p\equiv4,7,8\pmod 9$, so that rank $1$ is expected there. A manuscript of the OpenAI Math Release states rank one for those primes; the problem page records it as context with no verification.

Depends on. No page of this wiki; the proof is the paper's own.

Acceptance. Refereed: Rad Hrvatske akademije znanosti i umjetnosti, Matematičke znanosti, volume 29 (2024), published by the Croatian Academy of Sciences and Arts (the Crossref record). The site's curator, Thomas F. Bloom, lists Walsh's construction in the problem page's remarks, where the label is OPEN; that is not acceptance, so no reviewed is listed. The page is named by the claim's first posting, arXiv:2404.03970v1 of 5 April 2024.