Solution to the Erdos problem on distinct residues of factorials

TL;DR

This paper provides an elementary proof that no prime greater than 5 exists for which the residues of factorials from 2! to (p-1)! are all distinct modulo p, answering Erdos's question.

Contribution

It offers a simple, elementary proof that such a prime p does not exist, resolving Erdos's problem.

Findings

No prime p > 5 has factorial residues 2! to (p-1)! all distinct modulo p.

The proof is elementary and does not rely on advanced number theory.

Erdos's question is definitively answered in the negative.

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.