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Claim. The integers $12167=23^3$ and $12168=2^3\cdot3^2\cdot13^2$ are consecutive powerful numbers and neither is a square, since $110^2=12100<12167<12168<12321=111^2$. So the first question of Problem 365, read as the site reads it (must one of $n$, $n+1$ be a square?), has answer no. The observation is in S. W. Golomb, Powerful numbers, Amer. Math. Monthly 77 (1970), no. 8, 848--852 (the site's reference list prints 848--855). Golomb splits consecutive powerful pairs into those with a square member and those without, the division Walker later described completely.

Covers. The first question only (the part pell): one counterexample refutes it. The second question, whether the number of such $n\le x$ is $(\log x)^{O(1)}$, is untouched.

Depends on. Nothing in this wiki; the claim is a finite check recorded in the cited paper.

Acceptance. Refereed: the paper appeared in the American Mathematical Monthly, a refereed journal, in October 1970, the month this page is dated to. The site's commentary records the answer no and credits Golomb, but the site labels the problem OPEN (page last edited 31 October 2025), so that commentary is not acceptance of the problem and the page lists no reviewed evidence. Formalization: Collin Yuanjie Ren's Lean package JSP-000301, pinned above at the commit of 2026-09-16, proves consecutive_powerful_not_always_square, that not every pair of consecutive powerful positive integers has a square member, through the explicit witness $12167$, $12168$; its README credits the counterexample to Golomb, says the contribution sought is formalization rather than a new result, and says the code was prepared with OpenAI Codex assistance, with the axioms propext, Classical.choice and Quot.sound. The corpus has not built or audited it, so no formalized evidence is listed. The arithmetic of the counterexample is checked above; nothing else is reviewed here.