Kurepa's hypothesis asserts that for each integer n≥2 the greatest
common divisor of !n:=∑k=0n−1k! 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 Kp of order p−4
by a prime p≥7. In the previous version of this article we have proposed
the strong Kurepa's hypothesis involving a general Kurepa's determinant Kn
with any integer n≥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 Kn involving the derangement number Sn−1 with an odd
integer n≥9, we find that the integer 11563=31×373 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.Comment: 18 pages. This is the previous (first) version of the article
extended with Section 4 where we disprove the "odd composite part''of Strong
Kurepa's hypothesi