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Claim. K. Chen and S. Korsky, Improved Bounds for Distinct Multiples in Intervals, arXiv:2607.26450. Let $F(n)$ be the least $H$ such that any $H$ consecutive integers contain distinct $a_1,\ldots,a_n$ with $k\mid a_k$, which is $\max_mf(n,m)$ of Problem 711 up to the convention for the interval. The first version (29 July 2026), by Chen alone, gives $F(n)\ge n\exp(\tfrac1{50}\log n/\log\log n)$ for large $n$ and $F(n)\le n^{\beta+o(1)}\ll n^{1.4031}$, with $\beta\in(1,2)$ the root of $2\beta^3-8\beta^2+8\beta-1=0$. In the second version (13 August 2026), by Chen and Korsky, Theorem 1.1 gives $F(n)\le n^{4/3}\exp(O(\log n/\log\log n))$ and Theorem 1.3 gives

$$F(n)\ge h_{\mathbb P}(n) \ge n\exp\Bigl(\bigl(\tfrac{\log2}2-o(1)\bigr)\frac{\log n}{\log\log n}\Bigr),$$

where $h_{\mathbb P}(n)$ is the analogue of $F(n)$ for the primes up to $n$. The authors say that the lower bounds answer Kominers's question whether $F(n)\ll n\log n$ in the negative. The upper bound rests on a new estimate for unions of arithmetic progressions; the lower bound adapts a quadratic-residue construction of Green and Ruzsa. The first version states that its main proofs were developed with the assistance of ChatGPT 5.6 Sol, and the second that the authors used ChatGPT-5.6 Sol as an exploratory and proof-auditing tool.

Submission note. Posted to the site's forum by Samuel Korsky on 14 September 2026:

Kaizhe’s and my paper https://arxiv.org/pdf/2607.26450 immediately implies the improved lower bound $f(n) \ge \exp\left(\left(\frac{\log 2}{2}-o(1)\right)\frac{\log n}{\log\log n}\right)$ simply using $A=\{2,3,\ldots,n+1\}$.

GPT Astra claims that the argument of Lemma 2.1 in our paper can also be extended (1) using the GCD-sum estimate

$$\sum_{a,b\in > D}\frac{\gcd(a,b)}{\sqrt{ab}}\le |D|^{1+o(1)}.$$

found here:

https://arxiv.org/abs/1402.0249, and (2) an additional counting argument, to give $ f(n) \le n^{1/3+o(1)}.$ I have not yet reviewed that argument in detail, but can review and post if there's interest.

Covers. The second question, through a one-line deduction made on this page and not stated in the paper: Theorem 1.3, with Theorem 3 of Erdős and Pomerance (1980), $f(n,n)\ll n\sqrt{\log n}$, gives $F(n)-f(n,n)\to\infty$. It does not cover the first question, since the bound $n^{4/3+o(1)}$ settles no instance of it.

Standing. Pending. There is no journal reference, and the site has not credited the preprint.

Depends on. Erdős and Pomerance (1980), Theorem 3.