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Claim. David Turturean's write-up, an Overleaf document linked from their post of 2026-05-03 in the problem's discussion thread, claims that the abc conjecture implies, for all sufficiently large $n$,

$$n-O(\log n)\le f(n)\le n-2,$$

where $f(n)$ is the least $m$ with $n!=a_1\cdots a_t$ and $a_1<\cdots<a_t=a_1+m$, as in Problem 393. The upper bound is the factorization $n!=2\cdot3\cdots n$. The lower bound starts from Terence Tao's sketches in the same thread of 2025-09-16 and 2025-09-17: if $f(n)<n-C\log n$ for a large $C$, Kummer's theorem on the power of $2$ in $n!$ forces one factor to carry a power of $2$ of size at least $n^{10}$, so every factor is that large and, by Stirling's formula, there are only $O(\log n)$ factors; then for each prime $p\le n^{0.99}$ one factor carries almost all of $p$'s valuation in $n!$, and the pigeonhole principle gives two factors $a_i=a_j+O(n)$ each divisible by a product of prime powers of size about $a_i^{0.95}$, whose radicals are then far too small for the abc conjecture. The post says the write-up settles the problem partially under abc and that every attempt to remove the hypothesis left a kernel resembling abc itself.

Submission note. Posted to the site's forum by David Turturean on 3 May 2026:

Starting with the ideas of Tao in this thread, I put together a write-up that, conditional on abc, settles f(n) between the bounds: n - O(log n) $\leq$ f(n) $\leq$ n - 2 for all sufficiently large n.

The writeup is at this Overleaf link.

I tried to push this to an abc-unconditional proof, but every reduction I attempted and every sub-case I treated still left a remaining kernel that looks a lot like abc itself: typically a primitive triple of nearby integers with anomalously small radical relative to height, or an arithmetic configuration that morally invokes the same bound. In this sense, the problem is settled at least partially for now, and the abc-unconditional question remains hard.

The proof was developed via an automated multi-turn audit-and-revise scaffold that I built, which iteratively queried ChatGPT-5.5-Pro, running for tens of consecutive turns/prompts before the abc-conditional bound was first settled. (The scaffold continued running for long afterward in unsuccessful attempts to remove the abc dependence.)

Here is ChatGPT-5.5-Pro verifying the solution: check 1, check 2, check 3.

Hypothesis. The claim is conditional on the abc conjecture: for every $\varepsilon>0$ there is a constant $C(\varepsilon)$ such that coprime positive integers $a+b=c$ satisfy $c<C(\varepsilon)\,\mathrm{rad}(abc)^{1+\varepsilon}$. The conjecture is unproved, so this page derives nothing for the problem's standing; acceptance would establish the implication only. Under the same hypothesis, Luca 2002 already gives $f(n)\to\infty$.

Authorship and system. The post says the proof was developed by an automated multi-turn audit-and-revise scaffold that the author built, which queried ChatGPT-5.5-Pro over tens of consecutive turns before the abc-conditional bound was first obtained, and it links three ChatGPT-5.5-Pro checks of the solution. The submitter is the claimant.

Standing. The write-up is not refereed, is not on arXiv and no outside reviewer has recorded accepting it; a commenter wrote on 2026-05-04 that a standard check found no issues in the write-up, which is not acceptance. The site labels the problem OPEN and its remarks do not mention the write-up. The claim is claimed.

Depends on. No page of this wiki.