A new diophantine equation involving fifth powers

TL;DR

This paper presents a parametric solution to a previously unsolved fifth-power Diophantine equation and demonstrates, through elliptic curves, the existence of infinitely many such solutions that can be explicitly calculated.

Contribution

It introduces a novel parametric solution to a complex fifth-power Diophantine equation and proves the infinite nature of solutions using elliptic curve techniques.

Findings

Existence of infinitely many solutions

Explicit parametric solutions can be computed

Application of elliptic curves to Diophantine equations

Abstract

In this paper we obtain a parametric solution of the hitherto unsolved diophantine equation $(x_1^5+x_2^5)(x_3^5+x_4^5)=(y_1^5+y_2^5)(y_3^5+y_4^5)$. Further, we show, using elliptic curves, that there exist infinitely many parametric solutions of the aforementioned diophantine equation, and they can be effectively computed.