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Claim. Let $F(n)$ be the largest prime factor of $n(n+1)$, the quantity of Problem 368. For every $\varepsilon>0$ and all sufficiently large $n$,

$$F(n)>\frac{\log\log n}{1+\varepsilon},$$

so $F(n)\gg\log\log n$. This follows from the theorem on p. 4 of K. Mahler, Über den grössten Primteiler spezieller Polynome zweiten Grades, Archiv for Mathematik og Naturvidenskab (1935), recorded on the library card Mahler 1935: if $A_0$ is one of $\pm1,\pm2$, $D_1$ is squarefree and coprime to $A_0$, and $x_0$ is a natural number coprime to $A_0$, then for every $\varepsilon>0$ and all large $x_0$ the number $D_1x_0^2-A_0$ has a prime factor $p>(\log\log x_0)/(1+\varepsilon)$. With $D_1=1$, $A_0=1$ and $x_0=2n+1$ one has $x_0^2-1=4n(n+1)$; the prime factor the theorem gives exceeds $2$ once $n$ is large, so it divides $n(n+1)$, and $\log\log(2n+1)\ge\log\log n$. The method extends Størmer's theory of the solutions of $x^2-Dy^2=1$ whose $y$ has all its prime factors dividing $D$, as the card records.

Covers. The lower bound $F(n)\gg\log\log n$ only. It improves Pólya's $F(n)\to\infty$ and is superseded by Pasten's $(\log\log n)^2/\log\log\log n$.

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

Acceptance. Refereed: the paper appeared in the journal Archiv for Mathematik og Naturvidenskab in 1935; the record gives no month, so the page is dated to the first day of the publication year. The site's commentary credits Mahler with the bound, but the site labels the problem OPEN, so that commentary is not acceptance and the page lists no reviewed evidence. The linked page is the collected-works archive hosting the paper. The proof is not reviewed in this corpus.