The Kurepa-Vandermonde matrices arising from Kurepa's left factorial hypothesis

Abstract

Kurepa's (left factorial) hypothesis asserts that

for each integer $n\ge 2$ the greatest common divisor of

$!n:=\sum_{k=0}^{n-1}k!$ and $n!$ is $2$.

It is known that Kurepa's hypothesis is equivalent to

  $$

\sum_{k=0}^{p-1}\frac{(-1)^k}{k!}\not\equiv 0\pmod{p}\quad

{\rm for\,\, each\,\,odd\,\, prime}\quad p,

  $$

or equivalently, 

$S_{p-1}\not\equiv 0(\bmod{\,p})$

(i.e., $B_{p-1}\not\equiv 1(\bmod{\,p})$)

for  each odd prime $p$,

where $S_{p-1}$ and $B_{p-1}$ are the $(p-1)$th derangement number

and the $(p-1)$th Bell number, respectively.

Motivated by these two reformulations of Kurepa's hypothesis

and a congruence involving the Bell numbers  and the

derangement numbers established 

by  Z.-W. Sun and D. Zagier \cite[Theorem 1.1]{sz},

here we give two ``matrix'' formulations  of Kurepa's hypothesis

over the field $\Bbb F_p$, where $p$ is any odd prime.

The matrices $V_p$ and $C_p$ which are involved in these 

``matrix'' formulations of Kurepa's hypothesis  are  the

square $(p-1)\times (p-1)$ Vandermonde-like matrices.

Accordingly, $V_p$ and $C_p$

are called the Kurepa-Vandermonde matrices.

Furthermore, for each odd prime $p$ we determine

$\det (V_p)$ and $\det (C_p)$ in the field $\Bbb F_p$.