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Claim. Let $h(n)$ be as in Problem 860 and let $F(n)$ be the function of Problem 711, the least $H$ such that any $H$ consecutive integers contain distinct multiples of $1,2,\ldots,n$, the $k$-th divisible by $k$. Kaizhe Chen filed the claim on the site's proof-claims tab on 29 July 2026, marked as made with the AI system ChatGPT 5.6 Sol: a lower bound $h(n)\ge n\exp(c\log n/\log\log n)$ for some constant $c>0$, and the upper bounds $F(n)\ll n^{1.4031}$ and $h(n)\ll n^{1.4}$. The tab entry links no write-up; it says the results were obtained about three weeks before the filing, that a polished version had been submitted to arXiv, and proposes merging with Korsky's claim.

The arXiv paper is Improved Bounds for Distinct Multiples in Intervals, arXiv:2607.26450, whose $h_{\mathbb P}(n)$ is this problem's $h(n)$ less one (the site's open interval holds $h(n)-1$ integers), a shift no stated bound feels. Its first version (29 July 2026, by Kaizhe Chen alone) states the tab's bounds: $F(n)\le n^{\beta+o(1)}\ll n^{1.4031}$ with $\beta$ the root in $(1,2)$ of $2\beta^3-8\beta^2+8\beta-1=0$, $h_{\mathbb P}(n)\ll n^{7/5}/(\log n)^{2/5}$, and $F(n)\ge h_{\mathbb P}(n)\ge n\exp(\frac1{50}\frac{\log n}{\log\log n})$ for large $n$. Its second version (13 August 2026, joint with Samuel Korsky, arXiv comment "Improved lower and upper bounds") sharpens them: Theorem 1.1 gives $F(n)\le n^{4/3}\exp(O(\log n/\log\log n))$, Theorem 1.2 gives $h_{\mathbb P}(n)\ll n^{4/3}/(\log n)^{1/3}$, and Theorem 1.3 gives $F(n)\ge h_{\mathbb P}(n)\ge n\exp((\frac{\log2}2-o(1))\frac{\log n}{\log\log n})$. The upper bounds rest on a new estimate for unions of arithmetic progressions; the lower bound adapts a quadratic-residue compression construction of Green and Ruzsa, and the second version's constant is the result of Korsky's earlier claim, whose page records it. The second version's statement on AI says the authors used ChatGPT-5.6 Sol as an exploratory and proof-auditing tool.

Submission note. Posted to erdosproblems.com as a proof claim by Kaizhe Chen (account Kaizhe) on 29 July 2026, giving "ChatGPT 5.6 Sol" as the AI used:

Using ChatGPT 5.6 Sol, we proved $h(n)\ge n\exp\left(\frac{c\log n}{\log\log n}\right)$ for some constant $c$. Moreover, we proved new upper bounds $F(n)\ll n^{1.4031}$ and $h(n)\ll n^{1.4}$, where $F(n)$ is the function defined in Problem 711. The results are obtained about three weeks ago, and a fully polished version has been submitted to arXiv. I suggest merging the paper, but I don't know the email address of SamKorsky.

Covers. The upper bound $h(n)\ll n^{1.4}$ of the tab and the first version, and a lower bound of the shape $h(n)\ge n\exp(c\log n/\log\log n)$ with an unspecified constant; the sharper constant $\frac{\log2}2$ is Korsky's result, and the second version's $h(n)\ll n^{4/3}/(\log n)^{1/3}$ and $F(n)\le n^{4/3}\exp(O(\log n/\log\log n))$ are the joint results of Chen and Korsky, Theorems 1.1 and 1.2 of that version, recorded on their joint page. The tab's $h(n)\ll n^{1.4}$ improves the bound $h(n)\ll n^{3/2}/(\log n)^{1/2}$ of Erdős and Pomerance, which the site's commentary records as the known upper bound. With the lower bound, the two versions together place $h(n)$ between $n^{1+o(1)}$ and $n^{4/3}$; the order of magnitude the problem asks for stays open.

Standing. The tab lists the claim as a partial proof with one comment and warns that listing is no check of correctness; the site's label is unchanged and its commentary does not record the bounds. The arXiv paper has no journal reference, and no reviewer independent of the claimants has endorsed the argument. The claim stays claimed.

Depends on. Nothing on this wiki; the arguments are the paper's own.