Erdős Problem #976

The problem

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 $\gg n^d$?

Statement quoted from erdosproblems.com/976, maintained by Thomas Bloom. Original source: [Er65b,p.217]. Current status there: open.

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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