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Claim. Assume the Elliott–Halberstam conjecture for friable integers. Then the set of $n$ with $P(n)<P(n+1)$, where $P$ is the largest prime factor, has asymptotic density $1/2$, the statement of Problem 371. The paper's abstract states this as one of three conjectures on $P(n)$ and $P(n+1)$ that follow from the hypothesis, the others being the joint Dickman law of Problem 928, that the $n\le x$ with $P(n)\le x^{s}$ and $P(n+1)\le x^{t}$ have density $\rho(1/s)\rho(1/t)$ for $s,t\in(0,1)$, and 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). The site's commentary (page last edited 23 January 2026) credits the paper with the asymptotic density and in particular with a conditional affirmative answer to the question.

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 $x^{1/2-\epsilon}$. The library holds no copy of the paper, and 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 natural density was known in weaker senses only, as the problem page records, until the OpenAI release claimed it 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; the hypothesis is stated above.