Improved algorithms for left factorial residues

Vladica Andrejić (1) , Alin Bostan (2) , Milos Tatarevic (3)

1 University of Belgrade [Belgrade]
2 SPECFUN - Symbolic Special Functions : Fast and Certified
3 Chercheur indépendant

We present improved algorithms for computing the left factorial residues $!p=0!+1!+\dots+(p-1)! \!\mod p$. 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

Mots clés: Kurepa's conjecture; Left factorial; Socialist primes; Fast algorithms; Software design and implementation

Domaines: Calcul formel [cs.SC]; Théorie des nombres [math.NT]

Vladica Andrejić, Alin Bostan, Milos Tatarevic. Improved algorithms for left factorial residues. Information Processing Letters, 2021, 167, pp.3. ⟨10.1016/j.ipl.2020.106078⟩. ⟨hal-02411741⟩