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

DOI:
https://doi.org/10.1090/mcom/3098

Published electronically:
March 24, 2016

MathSciNet review:
3522982

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Abstract | References | Similar Articles | Additional Information

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

Email:
andrew@matf.bg.ac.rs

**Milos Tatarevic**

Affiliation:
Alameda, California 94501

Email:
milos.tatarevic@gmail.com

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