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Problem 372 — Erdős Problems

Problem 372

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

Source: erdosproblems.com/372

Number theory

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Solved·, Proved

An accepted solution exists. The statement is true.

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.

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.

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Status: Solved, Proved
Topics: Number theory
In the wiki: Wiki page