Last modified: 2014-01-29
Abstract
\begin{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
$$
\sum_{k=0}^{p-1}\frac{(-1)^k}{k!}\not\equiv 0\pmod{p}\quad
{\rm for\,\, each\,\, prime}\quad p\ge 3.
$$
Motivated by the above reformulation of Kurepa's hypothesis,
and using a Linear Algebra approach, for every integer $n\ge 7$
we define the so called Kurepa's determinant $K_n$ of order $n-4$.
We prove that Kurepa's hypothesis is equivalent
with the assertion that $K_p\not\equiv 0(\bmod{\,p})$ for all
primes $p\ge 7$. Furthermore, we establish some interesting
divisibility properties of Kurepa's determinants $K_n$ with
composite integers $n\ge 8$.
Related computational results are also presented.
\end{abstract}