Erdős (Erdos) Problem #976: status in 2026

Erdős Problem #976

Let f∈ ℤ[x] be an irreducible polynomial of degree d≥ 2. Let F_f(n) be maximal such that there exists 1≤ m≤ n with f(m) is divisible by a prime ≥ F_f(n). Equivalently, F_f(n) is the greatest prime divisor of Π_1≤ m≤ nf(m). Estimate F_f(n).

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-08-31).

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #976 as open (status updated 31 August 2025).

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #976 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: research statements in the file	3 open, 3 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): erdosproblems.com (checked 2026-11-12), Formal Conjectures (checked 2026-11-12). How it works.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite

@misc{cairn-erdos-976,
  title        = {Erdős Problem #976},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-976}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

Verified

Disputed

Refuted

On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

erdos_976. Let $f \in Z [x]$ 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 $1 \leq m \leq n \prod f (m) .$ Estimate $F_{f} (n)$ .

We read "estimate" as determining the order of magnitude of $F_{f} (n)$ .

erdos_976.parts.i. Let $f \in Z [x]$ be an irreducible polynomial of degree $d \geq 2$ , and let $F_{f} (n)$ be the greatest prime divisor of $\prod_{1 \leq m \leq n} f (m)$ . Is it true that $F_{f} (n) ≫ n^{1 + c}$ for some constant $c > 0$ ?

erdos_976.parts.ii. Let $f \in Z [x]$ be an irreducible polynomial of degree $d \geq 2$ , and let $F_{f} (n)$ be the greatest prime divisor of $\prod_{1 \leq m \leq n} f (m)$ . Is it true that $F_{f} (n) ≫ n^{d}$ ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«976» (3 statements).

theorem erdos_976 (f : ℤ[X]) (hf : Irreducible f) (hd : 2 ≤ f.natDegree) :
    (fun n ↦ (F f n : ℝ)) =Θ[atTop] erdos_976.answer f

theorem erdos_976.parts.i :
    ∀ f : ℤ[X], Irreducible f → 2 ≤ f.natDegree →
      ∃ c > (0 : ℝ), (fun n : ℕ ↦ (n : ℝ) ^ (1 + c)) =O[atTop] fun n ↦ (F f n : ℝ)

theorem erdos_976.parts.ii :
    ∀ f : ℤ[X], Irreducible f → 2 ≤ f.natDegree →
      (fun n : ℕ ↦ (n : ℝ) ^ f.natDegree) =O[atTop] fun n ↦ (F f n : ℝ)

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already — check erdosproblems.com/976. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/976
[Er52c] Erdős, P., On the greatest prime factor of $\prod_{k = 1}^{x} f (k)$ . J. London Math. Soc. 27 (1952), 379-384.
[Er65b] Erdős, P., Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244.
[Na22] Nagell, T., Abh. Math. Sem. Hamburg 1 (1922), 179-194.
[Ri34] Ricci, G., Annali di Mat. (1934), 295-303.
[Te90] Tenenbaum, G., Sur une question d'Erdős et Schinzel. A tribute to Paul Erdős (1990), 405-443.

Source and licence

Imported from Formal Conjectures (Erdős problems), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.