  In this paper we study the Diophantine equation $x^{4}-q^{4}=py^{5},$ with
the following conditions: $p$ and $q$ are different prime natural numbers, $y$
is not divisible with $p$, $p\equiv3$ (mod20), $q\equiv4$ (mod5),
$\overline{p}$ is a generator of the group $(U(\textbf{Z}_{q^{4}}),\cdot)$,
$(x,y)=1$, 2 is a 5-power residue mod $q$.
