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Claim. Assume the Elliott–Halberstam conjecture for friable integers. Then for every $s,t\in(0,1)$ the set $\{n\le x: P^+(n)\le x^{s},\ P^+(n+1)\le x^{t}\}$ has asymptotic density $\rho(1/s)\rho(1/t)$, where $P^+$ is the largest prime factor and $\rho$ is Dickman's function: the two smoothness events are independent in the sense Erdős asked for in Problem 928. The paper's abstract states this as the first of three conjectures on $P^+(n)$ and $P^+(n+1)$ that follow from the hypothesis, the one it attributes to Erdős and Pomerance; the others are that for every $\alpha$ the $n$ with $P^+(n+1)>P^+(n)n^{\alpha}$ have an asymptotic density, equal to the value Teräväinen had found for the logarithmic density (Teräväinen 2018), and that the $n$ with $P^+(n)<P^+(n+1)$ have density $1/2$, the question of Problem 371, recorded on its own claim page. The site's commentary (page last edited 3 April 2026) credits the paper with the density $\rho(1/\alpha)\rho(1/\beta)$ under the hypothesis.

Formulation. The problem's set uses the thresholds $n^{\alpha}$ and $(n+1)^{\beta}$ with strict inequalities; Wang's set uses the fixed thresholds $x^{s}$ and $x^{t}$ with weak inequalities. The strict and weak forms differ, up to $X$, by at most $\pi(X^{\alpha})+\pi((X+1)^{\beta})$ integers, the density-zero bridge stated in the problem page's Formulation. The fixed thresholds pass to the thresholds $n^{\alpha}$ and $n^{\beta}$ by a dyadic decomposition: for $x/2<n\le x$ the threshold $n^{\alpha}$ lies between $(x/2)^{\alpha}$ and $x^{\alpha}$, so each block's count is sandwiched between fixed-threshold counts at exponents tending to $\alpha$ and $\beta$, and the continuity of $\rho$ closes the gap. Neither bridge is in the paper; both are elementary and are stated here, not in Lean. The release's manuscript recorded on OpenAI 2026 treats the two threshold forms as separate statements and claims both without hypothesis.

Hypothesis. The Elliott–Halberstam conjecture for friable integers is a level-of-distribution hypothesis: the friable (smooth) integers up to $x$ are to be distributed in arithmetic progressions, on average over moduli up to a power $x^{1-\epsilon}$ of the range, as well as the Bombieri–Vinogradov theorem gives for moduli up to $x^{1/2-\epsilon}$. This page names the hypothesis as the paper's title and abstract and the site's commentary give it. It is unproved, and the claim gives no unconditional answer.

Scope. The claim is conditional and settles no standing of the problem by itself. Unconditionally, the product law was known in logarithmic density (Teräväinen 2018, cited above) and, for ordinary averages, outside an exceptional set of scales of logarithmic density zero (Tao and Teräväinen, Algebra Number Theory 13 (2019), Remark 3.3), until the OpenAI release claimed the natural density at every large scale without hypothesis on its claim page.

Acceptance. Published in J. Number Theory 223 (2021), 1–11, a refereed journal, DOI 10.1016/j.jnt.2020.12.013 (refereed); the publisher's record registers the article on 2021-01-20, which dates this page, and places it in the June 2021 issue. No arXiv posting is recorded. The site labels the problem OPEN, so no curator acceptance is listed.

Depends on. Nothing in this wiki beyond the problem page's Formulation, whose density-zero bridge is restated above; the hypothesis is stated above.