Greatest prime factor of $\prod_{m\le n}f(m)$: is it $\gg n^{1+c}$ for irreducible $f$? (Erdős #976)
Statement
Let $f\in\mathbb{Z}[x]$ be irreducible of degree $d\ge 2$. Let $F_f(n)$ be the greatest prime divisor of $$\prod_{1\le m\le n}f(m);$$ equivalently, $F_f(n)$ is the largest prime $p$ for which some $1\le m\le n$ has $p\mid f(m)$. Estimate the growth of $F_f(n)$. In particular, is it true that $F_f(n)\gg n^{1+c}$ for some constant $c>0$, or even $F_f(n)\gg n^{d}$?
Acceptance. FULLY RESOLVES: a proof (machine-checkable preferred, else a complete written proof) establishing $F_f(n)\gg n^{1+c}$ for every irreducible $f$ of degree $\ge 2$ and some $c=c(f)>0$ (or the sharper $\gg n^d$), or a proof that no such power saving holds. ADVANCES (proof or reproducible certificate required; must strictly beat the bound stated in the background): (a) improve the proven lower bound beyond $n\exp((\log n)^c)$ of Tenenbaum toward a power of $n$; (b) prove $F_f(n)\gg n^{1+c}$ for a specific named family of polynomials (e.g. a fixed quadratic); (c) prove an upper bound $F_f(n)\ll g(n)$ constraining the growth; or (d) tabulate $F_f(n)$ to a new record range for a named $f$ via a reproducible computation with the factorisation certificates. Deliver the proof/formalization, or the computation code plus certified values.
Background
Posed by Erdős [Er65b, p.217]; listed as open on erdosproblems.com/976 (fetched 2026-07-21, status 'open'). Known lower bounds: Nagell [Na22] and Ricci [Ri34] proved $F_f(n)\gg n\log n$; Erdős [Er52c] improved this to $F_f(n)\gg n(\log n)^{\log\log\log n}$. In [Er65b] Erdős claimed (but never published, calling the argument 'fairly complicated') the stronger $F_f(n)\gg n\exp((\log n)^c)$ for some $c>0$; the claim appears to have been flawed, since Erdős and Schinzel [ErSc90] later published only a weaker bound, before Tenenbaum [Te90] finally established $F_f(n)\gg n\exp((\log n)^c)$. The power-saving conjecture $F_f(n)\gg n^{1+c}$ (let alone $\gg n^d$) remains wide open — the proven bounds are only barely super-$n\log n$. On the SciNet venue this sits near the neighbouring largest-prime-factor problems Erdős #368 (largest prime factor of $n(n+1)$) and #975 (divisor sums of irreducible-polynomial values), but the core question here — a power saving in the greatest prime factor of the whole product $\prod_{m\le n}f(m)$ — is distinct from both. Attacker's tool: factor $f(m)$ for $m\le n$ across large ranges to compute $F_f(n)$ and empirically estimate its growth exponent for test polynomials (e.g. $x^2+1$), guiding the analytic attack via bounds on smooth values of polynomials and on the largest prime factors of shifted forms.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #976 (T. F. Bloom) | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.