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Claim. With $n_k$ the quantity of Problem 1063, there is an absolute constant $c>0$ such that

$$\log n_k\ge c(\log k)^2$$

for all $k$ (the tab's summary states no range, so the range is recorded as the summary gives it). Ricky Cipollini submitted this to the site's proof-claim tab on 4 August 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 note says the model wrote the paper from a modified version of a prompt in circulation. The write-up is the same read-only link on a collaborative editing service as the claimant's upper-bound claim; the service 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 4 August 2026, giving "GPT-5.6 Sol" as the AI used:

GPT 5.6-Sol proves a lower bound of $\log n_k\ge c(\log k)^2$. Notes: The paper was written by GPT 5.6-Sol using a slightly modified version of Liam Price's prompt.

Context. The lower bounds in the discussion thread before this claim are elementary: $n_k\ge2k$ by definition, and the claimant's thread comment of 26 June 2026 proves

$$n_k\ge\max\Bigl(2k,\ \prod_{p^a\parallel k}p^{\,a+\lfloor\log_p(k/2)\rfloor}\Bigr)$$

by locating the unique non-divisor $n-e$ through the $p$-adic valuation of $\binom nk$, which the comment of 10 July 2026 sharpens to the exponent $a+\lfloor\log_p(k-1)\rfloor$. Each prime $p$ dividing $k$ contributes a factor $p^{a+\lfloor\log_p(k/2)\rfloor}>p^{a-1}k/2\ge k/2$, so that bound is at least $(k/2)^{\omega(k)}$, with $\omega(k)$ the number of distinct prime factors of $k$: it is superpolynomial along $k$ with many prime factors, about $\exp\bigl((\log k)^2/\log\log k\bigr)$ along the primorials, but it equals $2k$ for prime $k$. The claimed bound $n_k\ge k^{c\log k}$ holds uniformly in $k$; its gain over the thread bound is largest for $k$ with few prime factors, and for $k$ with many prime factors it improves the exponent by at most a factor of order $\log\log k$. It remains far below the computed values, which grow exponentially in the range $k\le75$ of the thread's data; the claim does not assert sharpness.

Covers. A lower bound on $n_k$ superpolynomial in $k$. Not covered: any upper bound, the order of magnitude of $\log n_k$, and the estimate Erdős and Selfridge asked for. The same claimant's upper 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.