Improved algorithms for left factorial residues

2019/04/19 by Vladica Andrejić, Andrejić, Vladica, Alin Bostan +3
Computer Science · Mathematics · #11B83 (Primary) 11K31 (Secondary) #FOS: Computer and information sciences #FOS: Mathematics #Number Theory (math.NT) #Symbolic Computation (cs.SC) #cs.SC #math.NT #msc:11B83 #msc:11K31

paper · pdf · doi:10.48550/arxiv.1904.09196

final version, minor changes

arxiv created 2020/12/17 · arxiv updated 2020/12/21

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 2⁴⁰. Our results confirm that Kurepa's left factorial conjecture is still an open problem, as they show that there are no odd primes p<2⁴⁰ such that p divides !p. Additionally, we confirm that there are no socialist primes p with 5<p<2⁴⁰.

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