Variations of Kurepa's left factorial hypothesis

Abstract

Kurepa's hypothesis asserts that for each integer $n\ge 2$ the greatest common divisor of $!n:=\sum_{k=0}^{n-1}k!$ and $n!$ is $2$. Motivated by an equivalent formulation of this hypothesis involving derangement numbers, here we give a formulation of Kurepa's hypothesis in terms of divisibility of any Kurepa's determinant $K_p$ of order $p-4$ by a prime $p\ge 7$. In the previous version of this article we have proposed the strong Kurepa's hypothesis involving a general Kurepa's determinant $K_n$ with any integer $n\ge 7$. We prove the ``even part'' of this hypothesis which can be considered as a generalization of Kurepa's hypothesis. However, by using a congruence for $K_n$ involving the derangement number $S_{n-1}$ with an odd integer $n\ge 9$, we find that the integer $11563=31\times 373$ is a counterexample to the ``odd composite part'' of strong Kurepa's hypothesis. We also present some remarks, divisibility properties and computational results closely related to the questions on Kurepa's hypothesis involving derangement numbers and Bell numbers.