Erdős (Erdos) Problem #393: status in 2026

Erdős Problem #393

Let f(n) denote the minimal m≥ 1 such that n! = a_1⋯ a_t with a_1<⋯ <a_t=a_1+m. What is the behaviour of f(n)? We formalise the question as: does f(n) → ∞? A result of Luca [Lu02] implies a positive answer, conditional on the abc conjecture (see Erdos393.erdos_393.variants.luca).

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 #393 as open (status updated 31 August 2025).
Formal Conjectures marks the Lean statement erdos_393 as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #393 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: status of `erdos_393`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	2 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. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-erdos-393,
  title        = {Erdős Problem #393},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-393}},
  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

Nobody has worked on this problem here yet

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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

Let $f (n)$ denote the minimal $m \geq 1$ such that $n! = a_{1} \dots a_{t}$ with $a_{1} < \dots < a_{t} = a_{1} + m$ . What is the behaviour of $f (n)$ ?

We formalise the question as: does $f (n) \to \infty$ ? A result of Luca [Lu02] implies a positive answer, conditional on the $ab c$ conjecture (see Erdos393.erdos_393.variants.luca).

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«393».

theorem erdos_393 : Tendsto f atTop atTop

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
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/393. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

erdos_393.variants.infinitely_often_one — Erdős and Graham [ErGr80, p. 76] write that they do not even know whether f(n) = 1 for infinitely many n, that is, whether a factorial is the product of two…

References

erdosproblems.com/393
[ErGr80] Erdős, P. and Graham, R., *Old and new problems and results in combinatorial number theory*. Monographies de L'Enseignement Mathematique (1980).
[BeOs92] Berend, Daniel and Osgood, Charles F., *On the equation $P (x) = n!$ and a question of Erdős*. J. Number Theory 42 (1992), 189-193.
[BPZ23] Bui, Hung M. and Pratt, Kyle and Zaharescu, Alexandru, *Power savings for counting solutions to polynomial-factorial equations*. Adv. Math. 422 (2023), Paper No. 109021, 32 pp. arXiv:2204.08423
[Lu02] Luca, Florian, The Diophantine equation $P (x) = n!$ and a result of M. Overholt. Glas. Mat. Ser. III 37 (2002), 269-273.
OEIS A388302: the values $f (n)$ for $n \geq 2$ .

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.