Problem Statement
Let $f\in \mathbb{Z}[x]$ be an irreducible polynomial of degree $d\geq 2$. Let $F_f(n)$ be maximal such that there exists $1\leq m\leq n$ with $f(m)$ is divisible by a prime $\geq F_f(n)$. Equivalently, $F_f(n)$ is the greatest prime divisor of\[\prod_{1\leq m\leq n}f(m).\]Estimate $F_f(n)$. In particular, is it true that $F_f(n)\gg n^{1+c}$ for some constant $c>0$? Or even $\gg n^d$?
Categories:
Number Theory
Progress
Nagell and Ricci [Na22] proved that\[F_f(n) \gg n\log n,\]which Erdős [Er52c] improved to\[F_f(n) \gg n(\log n)^{\log\log\log n}.\]In [Er65b] he claimed a proof of\[F_f(n) \gg n\exp((\log n)^c)\]for some constant $c>0$, but said he had never published the proof, which was 'fairly complicated'. This seems to have been flawed, since Erdős and Schinzel [ErSc90] later published a weaker bound. A proof of the stronger bound above was finally provided by Tenenbaum [Te90].Source: erdosproblems.com/976 | Last verified: January 19, 2026