---

Claim. D. R. Heath-Brown, The largest prime factor of $X^3+2$, Proc. London Math. Soc. 82 (2001), 554--596, proves that for a positive proportion of the integers $n\in(X,2X]$ the largest prime factor of $n^3+2$ exceeds $X^{1+\varpi}$ with $\varpi=10^{-303}$, as Irving's 2015 paper on the same polynomial states the result. The paper's abstract gives the weaker form that the largest prime factor of $n^3+2$ is infinitely often at least $n^{1+\delta}$. For $f(t)=t^3+2$ the positive-proportion form gives $F_f(n)\geq(n/2)^{1+\varpi}$ for all large $n$, in the notation of Problem 976.

Covers. The first question for $f(t)=t^3+2$. It does not give $F_f(n)\gg n^3$.

Depends on. No page of this wiki.

Acceptance. Refereed: Proceedings of the London Mathematical Society 82 (2001), no. 3, whose issue the publisher's record dates May 2001; the page name uses the first day of that month. The site labels the problem OPEN.