Solution to the Erdos problem on distinct residues of factorials

Vyacheslav M. Abramov

Solution to the Erdos problem on distinct residues of factorials

Number Theory 2026-05-28 v3 Combinatorics

Authors: Vyacheslav M. Abramov

Abstract

Paul Erdos posed the following question: Is there a prime number $p>5$ such that the residues of $2!$ , $3!$ ,\ldots, $(p-1)!$ modulo $p$ all are distinct. In this short note, we give the negative answer on this question in an elementary way.

Keywords

Erdos problem Factorial residues Prime numbers Distinct residues

Cite

@article{arxiv.2604.26429,
  title  = {Solution to the Erdos problem on distinct residues of factorials},
  author = {Vyacheslav M. Abramov},
  journal= {arXiv preprint arXiv:2604.26429},
  year   = {2026}
}

Comments

9 pages. The presentation is shortened; unnecessary things are removed. Also, we added some small details that made things clearer. In general, the proof is the same as in the previous version

Solution to the Erdos problem on distinct residues of factorials

Abstract

Keywords

Cite

Comments

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