On the Diophantine equation x^4-q^4=py^5

Abstract

In this paper we study the Diophantine equation $x^4-q^4=p{y}^5,$ with the following conditions: $p$ and $q$ are different prime natural numbers, $y$ is not divisible with $p$, $p\equiv 3$ (mod20), $q\equiv 4$ (mod5), $\overline{p}$ is a generator of the group $(U({\text{Z}}_{q^4}),\cdot )$, $(x,y)=1$, 2 is a 5-power residue mod $q$.

This paper was accepted for publication in Italian Journal of Pure and Applied Mathematics