On distinct residues of factorials

TL;DR

This paper explores primes greater than 5 for which factorial residues are all distinct modulo p, linking the problem to Kurepa's function and providing computational evidence up to 10^11.

Contribution

It establishes a connection between factorial residue distinctness and Kurepa's function, and reports computational results on primes less than 10^11.

Findings

No such primes less than 10^11 found

Connection established between factorial residues and Kurepa's function

Provides computational evidence for the conjecture

Abstract

We investigate the existence of primes $p > 5$ for which the residues of $2!$, $3!$, \dots, $(p-1)!$ modulo $p$ are all distinct. We describe the connection between this problem and Kurepa's left factorial function, and report that there are no such primes less than $10^{11}$.