There are no socialist primes less than 10^9

TL;DR

This paper investigates primes for which factorials of all numbers less than the prime are distinct modulo that prime, providing evidence and conjecture about the non-existence of such primes below 10^9.

Contribution

The paper introduces a conjecture about the non-existence of primes with a specific factorial distinctness property below 10^9.

Findings

No such primes found below 10^9

Conjecture that no such primes exist at all

Provides computational evidence supporting the conjecture

Abstract

There are no primes $p$ with $5<p<10^{9}$ for which $2!, 3!, \ldots, (p-1)!$ are all distinct modulo $p$; it is conjectured that there are no such primes.