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Claim. Let $n_k\ge2k$ be the least $n$ such that $n-i$ divides $\binom nk$ for all but one $0\le i<k$, the quantity of Problem 1063. Then

$$\log n_k\le\frac{k}{\log k}\bigl(\log\log k+\log\log\log k+\log2+o(1)\bigr).$$

Ricky Cipollini submitted this to the site's proof-claim tab on 27 July 2026 (the page name's date), attributing the proof to the model GPT-5.6 Sol. The result is a bound rather than the estimate Erdős and Selfridge asked for, so the page records it as partial. The tab's summary says that the claimant first proved the weaker bound $\log n_k\le Ck\log\log k/\log k$ for an absolute constant $C>0$ and that the model sharpened it to the displayed form; the tab's note says the model wrote the paper from a modified version of a prompt in circulation. The write-up is a read-only link on a collaborative editing service that requires a sign-in, so this page records the claim from the tab's summary alone.

Submission note. Posted to erdosproblems.com as a proof claim by Ricky Cipollini (account rickyc) on 27 July 2026, giving "GPT-5.6 Sol" as the AI used:

GPT 5.6-Sol proves that $\log n_k\le \frac{k}{\log k}\bigl(\log\log k+\log\log\log k+\log 2+o(1)\bigr)$. This result comes from prompting it with a weaker bound I proved of $\log n_k \le C\,\frac{k\log\log k}{\log k}$ for some absolute constant $C>0$, which GPT-5.6 Sol improved to the bound proved in the paper. Notes: The paper was written by GPT 5.6-Sol using a slightly modified version of Liam Price's prompt.

Context. The upper bounds in the site's commentary before this claim are Monier's $n_k\le k!$ for $k\ge3$ (1985), which gives $\log n_k\le(1+o(1))\,k\log k$, and Cambie's sharpening $n_k\le k\,[2,3,\ldots,k-1]\le e^{(1+o(1))k}$, where $[\cdots]$ is the least common multiple, posted to the discussion thread on 1 October 2025 and adopted in the site's commentary, which improves this to $\log n_k\le(1+o(1))k$. The claimed bound divides the exponent of Cambie's bound by a factor of order $\log k/\log\log k$. The discussion thread's computed values of $n_k$, to $k=59$ in OEIS A389360 and to $k=75$ in a thread comment of 28 July 2026, are not compared with the bound here, since its $o(1)$ term fixes no finite check.

Covers. An upper bound on $n_k$: $\log n_k\le(1+o(1))\,k\log\log k/\log k$ with the stated second-order terms. Not covered: any lower bound, the order of magnitude of $\log n_k$, and the estimate Erdős and Selfridge asked for. The same claimant's lower bound is recorded on its own claim page; the two claims share one write-up link and are independent results.

Standing. Claimed. The write-up has no arXiv version and no journal record, the tab carried no comment under the claim on 2026-10-07, and the site's label is OPEN (page last edited 1 February 2026). Nothing here is this project's review, and no acceptance evidence is listed.

Depends on. No page of this wiki.