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

TL;DR

This paper investigates the solutions of the specific Diophantine equation x^4 - q^4 = p y^5 under certain prime and modular conditions, contributing to the understanding of fifth-power related equations.

Contribution

It provides new insights into the solutions of the equation with particular prime and residue conditions, expanding the knowledge of Diophantine equations involving fifth powers.

Findings

Conditions under which solutions exist are characterized.

Certain primes p and q lead to no solutions under the given constraints.

The structure of solutions is linked to properties of primitive roots and residues.

Abstract

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 $q$.