Searching for a counterexample to Kurepa's Conjecture

Abstract

Kurepa's conjecture states that there is no odd prime $p$ that divides $!p=0!+1!+\cdots +(p-1)!$. We search for a counterexample to this conjecture for all $p<2^{34}$. We introduce new optimization techniques and perform the computation using graphics processing units. Additionally, we consider the generalized Kurepa's left factorial given by ${!}^{k}n=(0!{)}^k+(1!{)}^k+\cdots +((n-1)!{)}^k$, and show that for all integers $1<k<100$ there exists an odd prime $p$ such that $p\mid {!}^{k}p$.

Accepted for publication in Mathematics of Computation

Cited by in corpus (2)

• On distinct residues of factorials
• Derangements and the $p$-adic incomplete gamma function