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Claim. Let $F(n)$ be the largest prime factor of $n(n+1)$, the quantity of Problem 368. There is an absolute constant $\kappa>0$ with

$$F(n)\ge\kappa\,\frac{(\log\log n)^2}{\log\log\log n}$$

for all large $n$. This is Corollary 1.5 of H. Pasten, The largest prime factor of $n^2+1$ and improvements on subexponential $ABC$, Invent. Math. 236 (2024), no. 1, 373--385 (arXiv:2312.03566, posted 2023-12-06, the date this page carries): for coprime positive integers $x<y$ the largest prime factor of $xy(x+y)$ is at least $\kappa(\log\log y)^2/\log\log\log y$, and the case $x=1$, $y=n$ is the bound stated. The paper's main theorem, the same bound for the largest prime factor of $n^2+1$, is on the library card Pasten 2024; the method combines linear forms in logarithms with the author's modular approach to the abc conjecture through Shimura curves. The bound improves Mahler's $\log\log n$ by nearly a square.

Covers. The lower bound $F(n)\gg(\log\log n)^2/\log\log\log n$ only. It does not determine the order of $F(n)$: the site expects $F(n)\gg(\log n)^2$, and the upper bounds along subsequences are on Schinzel's page.

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

Acceptance. Refereed: the paper appeared in Inventiones Mathematicae, a refereed journal, online on 2024-02-26. The site's commentary credits Pasten with the bound, but the site labels the problem OPEN, so that commentary is not acceptance and the page lists no reviewed evidence. The deduction from Corollary 1.5 is the one Boris Alexeev noted in the problem's thread on 2026-01-10, in answer to a question whether the paper, whose title names $n^2+1$, bears on $n(n+1)$. The proof is not reviewed in this corpus.