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Claim. S. D. Kominers, Long Intervals Without Distinct Multiples of the First $n$ Positive Integers, arXiv:2607.10431v1 (11 July 2026, 19 pages), announced by the author in the thread of Problem 711 the same day. With $f(n,m)$ the least $h\ge0$ such that $(m,m+h]$ holds distinct $a_1,\ldots,a_n$ with $i\mid a_i$, and $F(n)=\max_mf(n,m)$, Theorem 1.1 states

$$\liminf_{n\to\infty}\frac{F(n)-f(n,n)}{n\log n}\ge\frac1e,$$

so for each fixed $c<1/e$ and all large $n$ some interval of length $c\,n\log n$ holds no system of distinct multiples of $1,\ldots,n$. Corollary 1.2 gives $F(n)-f(n,n)>A\,n\log n/\log\log n$ for every fixed $A>0$ and all large $n$. The site's open interval adds one to every $f(n,m)$ and leaves the difference unchanged. The proof applies the Erdős–Pomerance smooth-number obstruction at starting points $m\asymp n\log n$, with a smoothness threshold of order $\log n$ and the local saddle-point estimates of Hildebrand and Tenenbaum.

Covers. The second question, answered yes at the scale $n\log n$; Remark 1.3 presents this as a strengthening of van Doorn's answer, on van Doorn's claim page. It does not cover the first question: the bound is consistent with $n^{1+o(1)}$, and Proposition 6.2, $\max_{m\le n^C}f(n,m)\le n^{1+o(1)}$ for each fixed $C>0$, reaches only starting points up to a fixed power of $n$, so it settles no instance of the first question.

Claimant and systems. The note's acknowledgments say that large language models assisted with computations, analysis, synthesis and verification, naming GPT-5.6 Sol, GPT-5.2 Pro, GPT-5.4 Pro, GPT-5.5 Pro and Claude Fable 5; that an exchange with GPT-5.6 Sol suggested changing the scale of the starting point; and that Refine.ink gave feedback on a prior draft. The problem, the final methods and the written form are the author's. The thread post credits GPT-5.6 Sol and Refine.ink.

Standing. Pending. There is no journal reference, and the site's commentary (last edited 11 January 2026) predates the note.

Depends on. No page of this wiki.