---

Claim. The second version of Improved Bounds for Distinct Multiples in Intervals, arXiv:2607.26450v2 (13 August 2026), by Kaizhe Chen and Samuel Korsky, with the arXiv comment "Improved lower and upper bounds". Its $h_{\mathbb P}(n)$ is the least $H$ for which every $H$ consecutive integers hold distinct multiples of the primes up to $n$, the $h(n)$ of Problem 860 less one (the site's open interval holds $h(n)-1$ integers). Theorem 1.2 states

$$h_{\mathbb P}(n)\ll\frac{n^{4/3}}{(\log n)^{1/3}},$$

and rests on a new local estimate for unions of arithmetic progressions, Lemma 2.1. Theorem 1.1 gives $F(n)\le n^{4/3}\exp(O(\log n/\log\log n))$ for the function $F(n)$ of Problem 711, the corresponding bound there. The paper's statement on AI says the authors used ChatGPT-5.6 Sol as an exploratory and proof-auditing tool. The same version's lower bound, Theorem 1.3, is Korsky's earlier claim; its first version, by Chen alone, is on Chen's page.

Covers. The upper bound $h(n)\ll n^{4/3}/(\log n)^{1/3}$, which improves Chen's $h(n)\ll n^{1.4}$ and the bound $h(n)\ll n^{3/2}/(\log n)^{1/2}$ of Erdős and Pomerance. The order of magnitude of $h(n)$ stays open.

Standing. The paper has no journal reference, and no reviewer independent of the authors has endorsed the argument. The claim stays claimed.

Depends on. Nothing on this wiki; the argument is the paper's own.