Problem 478

Let $p$ be a prime and

A_p = \{ k! \pmod{p} : 1\leq k<p\}.

Is it true that

\lvert A_p\rvert \sim (1-\tfrac{1}{e})p?

Source: erdosproblems.com/478

Number theory
Factorials

Wiki page

Status

Open

No claim settles this problem.

Open. The site labels the problem OPEN (page last edited 12 April 2026) and credits no solution; its remarks credit [GSSV24] with the best known lower bound $|A_p|\ge(\sqrt2-o(1))p^{1/2}$ . The standing derives from the claim pages: the accepted partial claim Grebennikov, Sagdeev, Semchankau and Vasilevskii proves that bound in a refereed paper, and the pending partial claim Hu 2026 is a manuscript with a partial Lean formalization proving $|A_p|\gg p^{8/15}$ . Neither reaches positive density nor the asymptotic asked for, so the problem is open with no settling or pending full claim.

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 $n! \bmod p$ . Rev. Mat. Iberoam. 40 (2024), no. 2, 637-648, doi:10.4171/rmi/1422.
Gu04: Guy, Richard K., Unsolved problems in number theory. Third edition, Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp. Section F11 "Distribution of residues of factorials", printed p. 381: the question of the distribution of $1!,\ldots,p!$ modulo $p$ , "About $p/e$ of the residue classes are not represented", the table of missing residues for $p\le37$ , and the Rokowska--Schinzel result. Library home: Guy 2004.
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. Erdős. Elem. Math. (1960), 84-85.
Tr13: T. Trudgian, There are no socialist primes less than $10^9$ . arXiv:1310.6403 (2013).

Problem 478

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Proof claims

Discussion