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Mathematics of Computation

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Searching for a counterexample to Kurepa’s conjecture

Authors: Vladica Andrejić and Milos Tatarevic
Journal: Math. Comp. 85 (2016), 3061-3068
MSC (2010): Primary 11B83; Secondary 11K31
Published electronically: March 24, 2016
MathSciNet review: 3522982
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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 $p\mid !^k p$.

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Additional Information

Vladica Andrejić
Affiliation: Faculty of Mathematics, University of Belgrade, Belgrade, Serbia
MR Author ID: 789950

Milos Tatarevic
Affiliation: Alameda, California 94501

Keywords: Left factorial, prime numbers, divisibility
Received by editor(s): September 2, 2014
Received by editor(s) in revised form: March 31, 2015, and June 17, 2015
Published electronically: March 24, 2016
Additional Notes: This work was partially supported by the Serbian Ministry of Education and Science, project No. 174012
Article copyright: © Copyright 2016 American Mathematical Society