Erdős (Erdos) Problem #478: status in 2026

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?

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 #478 as open (status updated 31 August 2025).
Formal Conjectures marks the Lean statement erdos_478 as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #478 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: status of `erdos_478`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	1 open, 0 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-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 2026-11-12}
}

Also: CITATION.cff · Atom feed of results

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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 $p$ be a prime and $A_{p} = {k! (mod p) : 1 \leq k < p} .$ Is it true that $∣ A_{p} ∣ \sim (1 - \frac{1}{e}) p ?$

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«478».

theorem erdos_478 :
    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 { $n! mod p$ }. 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 { $p$ }. 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 $1 0^{9}$ . arXiv:1310.6403 (2013).

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.