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Claim. With $P$ the largest prime factor, there are infinitely many $n$ with $P(n)<n^{1/2}$ and $P(n+1)<(n+1)^{1/2}$: take $n=m^2-1=(m-1)(m+1)$ with $m$ and $m+1$ composite. Every prime factor of $n$ divides $m-1$ or $m+1$ and so is at most $\max(m-1,(m+1)/2)=m-1<\sqrt{m^2-1}$ for $m\ge3$, and every prime factor of $n+1=m^2$ divides $m$ and so is at most $m/2<m=\sqrt{n+1}$. Consecutive composites $m,m+1$ exist beyond every bound, so this answers the question of Problem 370 yes. The site's commentary attributes the observation to Steinerberger and states it with $\ll$ in place of $<$, noting that choosing $m$ and $m+1$ composite gives the strict inequalities. The earliest dated record of the remark is a forum comment of 2025-10-17 that already refers to it, which dates this page; the site's revision history begins on 2025-10-20 with the remark present.

Acceptance. The site's curator, Thomas F. Bloom, records the construction in the problem's commentary as a trivial solution, labels the problem proved, and lists Steinerberger among those thanked on the page; Terence Tao's forum comment of 2025-10-17 accepts an argument of Steinerberger's type as valid and concludes that the problem was badly worded (reviewed). Two Lean 4 proofs of the formal-conjectures statement erdos_370 exist, and that project links both as formal proofs, which is the Lean that the site's label refers to. One was posted by Boris Alexeev on 2025-11-24 in the lean-proofs repository; its header declares it a formalization of a solution to the problem, names Steinerberger as the finder of the original proof, and says that a proof, not necessarily the original one, was explained by ChatGPT 5.1 Pro, auto-formalized by the Aristotle system and stated as in formal-conjectures; Alexeev wrote in the forum post of 2025-11-24 that they had checked the file and especially its final theorem statement. The other, by the GitHub user XC0R on 2026-04-13 in a fork of formal-conjectures, credits the trivial solution to Steinerberger in its docstring. Both therefore formalize this claim and are links on this page, not claims of their own; both take $m=j!+3$, so that $m-1$, $m$ and $m+1$ are composite for $j\ge3$. The statement file is linked above as a record. This corpus has not built either file or audited its statement, so formalized is not listed, and no refereed publication exists.

What the problem meant. The site's curator finds it strange that Erdős and Graham, who report Pomerance's observation on the problem, overlooked so simple a construction, suspects a misstatement, and offers no guess at the intended form. Erdős and Graham print Pomerance's remark (p. 69) as the statement that $P(n)>n^{e^{-1/2-\epsilon}}$, $P(n+1)>(n+1)^{e^{-1/2-\epsilon}}$ has solutions by density considerations, which is right, since integers with $P(n)>n^{e^{-1/2-\epsilon}}$ have density $1-\rho(e^{1/2+\epsilon})>1/2$. The site's paraphrase puts the exponent $1/\sqrt e-\epsilon$ into the problem's $<$ form instead, which is mistaken: $n^{c}$-smooth integers have density $\rho(1/c)$, above $1/2$ only for $c>1/\sqrt e$, so at the exponent $1/\sqrt e-\epsilon$ the density argument proves nothing. Tao's forum comment of 2025-10-17 reports a literature review with the Gemini and ChatGPT deep research tools: the regular Gemini LLM hallucinated a solution from the literature by Pell's equation, which Tao judged valid enough and similar to Steinerberger's, and suggested three variants (a set of $n$ of positive density, a smaller exponent, or longer runs of consecutive integers); the Gemini and ChatGPT deep research tools matched the variants to Problems 369 and 928. Tao concludes that the problem was badly worded and that those two problems capture its salvageable aspects.

Depends on. Nothing in this wiki.