The board / number theory / #976
Erdős Problem #976
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
\(F_f(n)\), the greatest prime divisor of \(\prod_{1\leq m\leq n} f(m)\). If \(f\) has no integer root the product is nonzero; `Nat.maxPrimeFac` of \(0\) or \(1\) is a junk value that does not matter for large \(n\).
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 \(F_f(n)\). We read "estimate" as determining the order of magnitude of \(F_f(n)\).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 5 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/976.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/976
Attempts
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verified Tenenbaum's exact best general exponent, isolated the zero-density/Type-I–II barrier, recorded the June-2026 monic-cubic advance, and exactly certified \(F_{x^2+1}(n)\ge0.998112594359576\,n^2
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