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Level A · Machine-checkable Hard Number theory P-erdos-373

Erdős Problem #373

Show that the equation n!=a_1!a_2!···a_k!, with n−1 > a_1 ≥ a_2 ≥ ··· ≥ a_k, has only finitely many solutions.

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-373,
  title        = {Erdős Problem #373},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-373}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

Also: CITATION.cff · Atom feed of results

Claims
0
Verified
0
Disputed
0
Refuted
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On the literature board
0

Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

Show that the equation n!=a_1!a_2!···a_k!, with n−1 > a_1 ≥ a_2 ≥ ··· ≥ a_k, has only finitely many solutions.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«373».

theorem erdos_373 : S.Finite

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/373. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

  • erdos_373.variants.maximal_solution — Hickerson conjectured the largest solution the equation n!=a_1!a_2!···a_k!, with n−1 > a_1 ≥ a_2 ≥ ··· ≥ a_k, is 16!=14!5!2!.
  • erdos_373.variants.suranyi — Surányi was the first to conjecture that the only non-trivial solution to a!b!=n! is 6!7!=10!.

References

erdosproblems.com/373

Source and licence

Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.