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Claim. With $P(n)$ the largest prime factor of $n$, there are infinitely many $n$ with $P(n)>P(n+1)>P(n+2)$; for all large $x$ the number of such $n\le x$ is $\gg\sqrt x$. This settles the statement of Problem 372, a conjecture of Erdős and Pomerance, who proved the ascending analog $P(n)<P(n+1)<P(n+2)$ for infinitely many $n$ and that $P(n)>P(n+1)$ holds on a set of positive lower density (Erdős and Pomerance 1978). Balog also conjectures that the $n$ with descending triples have density $1/6$; De Koninck and Doyon present a generalized form of that conjecture (De Koninck and Doyon 2011). The library holds no copy of Balog's paper; the statement above follows the site's commentary and the formal-conjectures file.

Acceptance. Published in Studia Sci. Math. Hungar. 38 (2001), 45–50, a refereed journal (refereed); the publisher's record dates the issue 2001-05-01, which dates this page. The site's curator, Thomas F. Bloom, records the problem as solved by this paper and labels it proved (reviewed). The formal-conjectures project states the problem as erdos_372 (category research solved) with no proof and links no formalization as of its 2026-09-18 commit, so formalized is not listed.

Depends on. Nothing in this wiki.