# Improved algorithms for left factorial residues

## TL;DR

This paper introduces more efficient algorithms for calculating left factorial residues, enabling computations for all primes up to 2^40, and confirms the conjecture's unresolved status within this range.

## Contribution

The authors develop improved algorithms for computing left factorial residues and apply them to verify the conjecture for all primes up to 2^40.

## Key findings

- No odd primes less than 2^40 divide !p
- No socialist primes with 5<p<2^40 found
- Conjecture remains open beyond 2^40

## Abstract

We present improved algorithms for computing the left factorial residues $!p=0!+1!+...+(p-1)! \!\mod p$. We use these algorithms for the calculation of the residues $!p\!\mod p$, for all primes $p$ up to $2^{40}$. Our results confirm that Kurepa's left factorial conjecture is still an open problem, as they show that there are no odd primes $p<2^{40}$ such that $p$ divides $!p$. Additionally, we confirm that there are no socialist primes $p$ with $5<p<2^{40}$.