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

Romeo Meštrović

Kurepa's (left factorial) hypothesis asserts that for each integer $n\geq2$ 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 \[ um_{k=0}^{p-1}\frac{(-1)^k}{k!}ot\equiv0\quad(peratorname{mod} p)\quadext{for each odd prime }p \] or equivalently, $S_{p-1}\not\equiv0\;(\operatorname{mod} p)$ (i.e., $B_{p-1}\not\equiv1\;(\operatorname{mod}p)$) 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 [28, Theorem 1.1], here we give two ``matrix'' formulations of Kurepa's hypothesis over the field $\mathbb F_p$, 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 $(p-1)\times(p-1)$ Vandermondelike matrices. Accordingly, $V_p$ and $C_p$ are called the Kurepa--Vandermonde matrices. Furthermore, for each odd prime $p$ we determine $\operatorname{det}(V_p)$ and $\operatorname{det}(C_p)$ in the field $\mathbb F_p$.