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Claim. Let $F(n)$ be the largest prime factor of $n(n+1)$, the quantity of Problem 368. For infinitely many $n$,

$$F(n)\le n^{O(1/\log\log\log n)}.$$

This follows from Theorem 14 of A. Schinzel, On two theorems of Gelfond and some of their applications, Acta Arith. 13 (1967), no. 2, 177--236 (the library card Schinzel 1967): for integers $A,E,r,s$ with $Ar\ne0$ and $E\mid4$, writing $q(m)$ for the greatest prime factor of $m$, the quantity

$$\frac{\log q(A(rx+s)^2-E)\,\log\log\log x}{\log\lvert A(rx+s)^2-E\rvert}$$

is bounded along infinitely many integers $x$ (the paper writes the conclusion as a finite limit as $x\to\infty$). With $A=E=1$, $r=2$ and $s=1$ one has $(2x+1)^2-1=4x(x+1)$, whose greatest prime factor is $F(x)$ once $x\ge2$, and $\log\lvert4x(x+1)\rvert\sim2\log x$; so $\log F(x)\le C\log x/\log\log\log x$ for infinitely many $x$, which is the bound. The paper's Theorem 13 gives the same conclusion for a general quadratic $A x^2-E$ along all $x$ but does not force $x$ odd, so Theorem 14, the improvement for $E\mid4$ along an arithmetic progression, is the exact source.

Covers. The upper bound along a subsequence only: $F(n)$ is at most $n^{O(1/\log\log\log n)}$ for infinitely many $n$. It does not bound $F(n)$ from below (the lower bounds are on the pages of Pólya, Mahler and Pasten) and it is far from Erdős's conjectured $(\log n)^{2+\varepsilon}$ infinitely often.

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

Acceptance. Refereed: the paper appeared in Acta Arithmetica, a refereed journal, in volume 13 (1967/68), issue 2; the record gives no month, so the page is dated to the first day of the volume's first year. The site's commentary credits Schinzel with the observation, but the site labels the problem OPEN, so that commentary is not acceptance and the page lists no reviewed evidence. The proof is not reviewed in this corpus.