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Level A · Machine-checkable Hard Number theory P-erdos-478

Erdős Problem #478

Let p be a prime and A_p = k! pmodp : 1≤ k<p. Is it true that lvert A_prvert ∼ (1-1/e)p?

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-478,
  title        = {Erdős Problem #478},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-478}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

Claims
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Verified
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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

Let be a prime and Is it true that

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«478». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_478 : answer(sorry) ↔
    Filter.Tendsto
      (fun p : ℕ =>
        (((Finset.Ico 1 p).image (fun k => Nat.factorial k % p)).card : ℝ) / p)
      (Filter.atTop ⊓ Filter.principal {p : ℕ | p.Prime})
      (nhds (1 - 1 / Real.exp 1))

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/478. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

  • erdosproblems.com/478
  • [AnTa16] V. Andrejić and M. Tatarevic, On distinct residues of factorials. arXiv:1603.04086 (2016).
  • [GSSV24] Grebennikov, Alexandr and Sagdeev, Arsenii and Semchankau, Aliaksei and Vasilevskii, Aliaksei, On the sequence {}. Rev. Mat. Iberoam. (2024), 637--648.
  • [Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.
  • [KlMu17] Klurman, Oleksiy and Munsch, Marc, Distribution of factorials modulo {}. J. Théor. Nombres Bordeaux (2017), 169--177.
  • [RoSc60] Rokowska, B. and Schinzel, A., Sur un problème de {M}. {E}rdős. Elem. Math. (1960), 84--85.
  • [Tr13] T. Trudgian, There are no socialist primes less than . arXiv:1310.6403 (2013).

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.