Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 7

Problem 478

claims/: The 2 claim pages of Problem 478, one per claimant's result; the problem's standing derives from them.

Statement. 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?

Status. 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.

Source. erdosproblems.com/478, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #478, https://www.erdosproblems.com/478.

References.

[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_unsolved_problems_number_theory.
[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).

Formalization. The formal-conjectures file FormalConjectures/ErdosProblems/478.lean, added on 2026-09-07, states the question as erdos_478, tagged open, with sorry and no formal proof, at the commit of 2026-09-18 linked here. The partial Lean development accompanying Hu's manuscript, which proves the $p^{8/15}$ bound from an unformalized incidence hypothesis of Stevens and de Zeeuw, is linked from the claim page at a pinned commit. This corpus has built neither.

Current assessment

Grebennikov, Sagdeev, Semchankau and Vasilevskii [GSSV24] prove $|A_p|\ge(\sqrt2+o(1))\sqrt p$ (claim page), and Hu's manuscript claims $|A_p|\gg p^{8/15}$ (claim page).

Klurman and Munsch [KlMu17] (J. Théor. Nombres Bordeaux 29 (2017), 169-177; card Klurman and Munsch 2017), whose results the site's remarks credit, have no claim page because none of them settles an instance of the asymptotic. Their Theorem 2.1 gives at least $\sqrt{3N/2}$ distinct values of $n!\bmod p$ for $H\le n\le H+N$ once $N\gg p^{1/4+\epsilon}$ . Their Theorem 3.1 shows that the mean of $p-|A_p|$ over primes $p\le x$ is $\gg\log\log x/\log\log\log x$ ; Theorem 3.2 raises this, under the Generalized Riemann Hypothesis, to $\gg x^{1/4}/\log x$ , and Corollary 3.3 deduces, under the same hypothesis, infinitely many primes with $p-|A_p|\gg p^{1/4}/\log p$ .

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (22)

A square-root lower bound with constant root two for factorial residues Hu's lower bound p to the 8/15 for distinct factorial residues problems/factorials_binomials/E0478/claims Factorials and Binomial Coefficients Factorials and Binomial Coefficients factorials_binomials/andrejic_2016_distinct_residues_factorials Computer search (pp. 2–3 and 5–6): there are no socialist primes p with 5 < p < 10^11 Congruences (2.4)–(2.6) (p. 2): a socialist prime has p ≡ 5 (mod 8), missing residue −((p−1)/2)!, and (!p − 2)^2 ≡ −1 (mod p)Condition (2.7) (p. 3): congruences for the generalized left factorial !^k p at a socialist prime Heuristic (4.1) (pp. 4–5): W_p ≤ (p−2)^{3/2} e^{3−p}, and at most about e^3 a^{3/2−a} √(ln a) socialist primes beyond a Section 3 (pp. 3–4): the involution (f(k))! ≡ −k!, its quadruples, and ((p−1)/4)! a quadratic residue at a socialist prime factorials_binomials/grebennikov_2024_sequence factorials_binomials/klurman_2017_distribution_factorials_modulo Corollary 3.3 (p. 10): under GRH, infinitely many primes p have p - V(0,p-1) >> p^(1/4)/log p Theorem 2.1 (p. 3): n! mod p takes at least sqrt(3N/2) values for H <= n <= H + N once N >> p^(1/4+eps)Theorem 3.1 (p. 6): on average over p <= x, n! mod p misses >> log log x / log log log x residue classes Theorem 3.2 (p. 9): under GRH, on average over p <= x, n! mod p misses >> x^(1/4)/log x residue classes factorials_binomials/trudgian_2013_there_are_no_socialist_primes_less Section 2 (p. 3): there are no socialist primes p with 5 < p < 10^9 Conditions (1) and (3) (pp. 1–3): necessary conditions for a socialist prime, including the new condition from the discriminant 1957 Conjecture (p. 4): there are no socialist primes, with a heuristic probability of about e^{(7−p)/2}number_theory/guy_2004_unsolved_problems_number_theory

Graph