  The title equation, where $p>3$ is a prime number $\not\equiv 7 \pmod 8$, $q$
is an odd prime number and $x,y,n$ are positive integers with $x,y$ relatively
prime, is studied. When $p\equiv 3\pmod 8$, we prove (Theorem 2.3) that there
are no solutions. For $p\not\equiv 3\pmod 8$ the treatment of the equation
turns out to be a difficult task. We focus our attention to $p=5$, by reason of
an article by F. Abu Muriefah, published in this journal, vol. 128 (2008),
1670-1675. Our main result concerning this special equation is Theorem 1.1,
whose proof is based on results around the Diophantine equation $5x^2-4=y^n$
(integer solutions), interesting in themselves, which are exposed in Sections 3
and 4. These last results are obtained by using tools such as Linear Forms in
Two Logarithms and Hypergeometric Series.
