On integer solutions to x^5 - (x+1)^5 -(x+2)^5 +(x+3)^5 = 5^m + 5^n

TL;DR

This paper presents an infinite family of integer solutions to a specific fifth-degree Diophantine equation involving powers of five, expanding understanding of solutions to such polynomial equations.

Contribution

It provides an infinite set of solutions to a particular fifth-degree Diophantine equation and explores solutions to related equations, advancing knowledge in number theory.

Findings

Infinite solutions to the given Diophantine equation.

Solutions to similar fifth-degree equations.

Enhanced understanding of power-based Diophantine equations.

Abstract

We give an infinite number of integer solutions to the Diophantine equation x^5 - (x+1)^5 -(x+2)^5 +(x+3)^5 = 5^m + 5^n, and some solutions to some similar equations.