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Claim. Let $F(n)$ be the largest prime factor of $n(n+1)$, the quantity of Problem 368. Then $F(n)\to\infty$ as $n\to\infty$: for every bound $P$ only finitely many $n$ have $n(n+1)$ composed of primes at most $P$. This is Satz I of G. Pólya, Zur arithmetischen Untersuchung der Polynome, Math. Z. 1 (1918), 143--148, applied to $f(x)=x(x+1)$: if $f$ is a product of two essentially different rational linear factors, the largest prime factor of $f(n)$ tends to infinity. The library card Pólya 1918 records the theorem and its proof's reduction, after fixing a finite prime set and taking exponents modulo $3$, to Thue's theorem on binary forms of degree at least three; the card also records that the theorem sharpens earlier results of Størmer, whose theory of the Pell equation gives the same conclusion for $n(n+1)$.

Covers. The qualitative statement $F(n)\to\infty$ only, with no rate. The quantitative lower bounds are on the pages of Mahler and Pasten.

Depends on. Nothing in this wiki; the claim rests on the cited paper.

Acceptance. Refereed: the paper appeared in Mathematische Zeitschrift, a refereed journal, in June 1918, the month this page is dated to. The site's commentary credits Pólya with the statement, but the site labels the problem OPEN, so that commentary is not acceptance and the page lists no reviewed evidence. Formalization: the file Erdos368b.lean in Boris Alexeev's lean-proofs repository, pinned above at the commit of 2026-02-17 and posted in the problem's thread that day, declares itself a formalization of a solution to a small part of the problem, credits the original proof to Pólya, and says that a proof of ChatGPT's choice, through Pell's equation, was auto-formalized by Aristotle from Harmonic; its n_n_plus_one_inf states that the largest prime factor of $n(n+1)$ tends to infinity. The corpus has not built or audited it, so no formalized evidence is listed. The proof is not reviewed in this corpus.