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Level A · Machine-checkable Hard Number theory P-paper-kurepa

Kurepa's conjecture

## Kurepa's conjecture For all n, !nnot≡ 0 mod n This appears as B44 "Sums of factorials." in Unsolved Problems in Number Theory by Richard K. Guy

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-paper-kurepa,
  title        = {Kurepa's conjecture},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/paper-kurepa}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Disputed
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Refuted
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On the literature board
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Current state

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

The problem

The question

Kurepa's conjecture

For all ,

This appears as B44 "Sums of factorials." in Unsolved Problems in Number Theory by Richard K. Guy

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Paper.Kurepa.

theorem kurepa_conjecture (n : ℕ) (h_n : 2 < n) : (!n : ℕ) % n ≠ 0

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

Variants

  • kurepa_conjecture.variants.prime — This statement can be reduced to the prime case only.
  • kurepa_conjecture.variants.gcd — An equivalent formulation in terms of the gcd of n! and !n.

References

On the left factorial function !N, by Đuro Kurepa Math. Balkanica 1, p. 147-153, 1971

Source and licence

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