Improved algorithms for left factorial residues

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 $5<p<2^{40}$.