Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 372

claims/: The 1 claim page of Problem 372, one per claimant's result; the problem's standing derives from them.

Statement. Let $P(n)$ denote the largest prime factor of $n$ . There are infinitely many $n$ such that $P(n)>P(n+1)>P(n+2)$ .

Status. PROVED, the site's label: solved by Balog [Ba01], who proves that there are $\gg\sqrt x$ such $n\le x$ for all large $x$ ; the statement is a conjecture of Erdős and Pomerance [ErPo78], and no Lean proof is linked. The accepted claim is Balog 2001.

Source. erdosproblems.com/372, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #372, https://www.erdosproblems.com/372.

References.

[Ba01] Balog, A., On triplets with descending largest prime factors. Studia Sci. Math. Hungar. (2001), 45-50.
[DeDo11] De Koninck, Jean-Marie and Doyon, Nicolas, On the distance between smooth numbers. Integers (2011), A25, 22.
[ErPo78] Erdős, Paul and Pomerance, Carl, On the largest prime factors of $n$ and $n+1$ . Aequationes Math. (1978), 311-321.

Formalization. Statement in formal-conjectures (erdos_372, category research solved, with no proof and no formal proof linked as of its 2026-09-18 commit).

Current assessment

Proved by Balog 2001, refereed. The site formulation above asserts infinitely many $n$ with $P(n)>P(n+1)>P(n+2)$ . Erdős and Pomerance conjectured it in 1978, proving the ascending analog and that $P(n)>P(n+1)$ has positive lower density, and Balog proved it in 2001 with $\gg\sqrt x$ such $n\le x$ for all large $x$ . Balog's conjectured density $1/6$ for the descending triples and its generalization by De Koninck and Doyon 2011 are open and are not the question. The library holds no copy of Balog's paper; the standing rests on the refereed publication and the site's record. No Lean proof is linked. Sources checked: the site's page and discussion thread (no comments), the formal-conjectures statement file at its 2026-09-18 commit, the publisher's record of [Ba01] and the library's cards for [ErPo78] and [DeDo11].

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (9)

Problem 371 Balog proves infinitely many descending triples problems/arithmetic_functions/E0372/claims Arithmetic Functions arithmetic_functions/dekoninck_2011_distance_between_smooth_numbers arithmetic_functions/erdos_1978_largest_prime_factors Construction (p. 320): P(n) < P(n+1) < P(n+2) for infinitely many n Corollary (p. 319): the n with P(n) > P(n+1) have lower density greater than 0.0099 Theorem (p. 320): the sum of ε_n/2^n is irrational, ε_n recording whether P(n) > P(n+1)

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