Perfect powers in sum of three fifth powers

TL;DR

This paper completely characterizes perfect powers that can be expressed as sums of three fifth powers in an arithmetic progression, solving a specific Diophantine equation involving such sums.

Contribution

It provides a complete solution to the Diophantine equation for sums of three fifth powers in arithmetic progression with specific coefficient constraints.

Findings

Identifies all solutions where the sum equals a perfect power

Classifies solutions based on the parameters d, x, and z

Advances understanding of sums of powers in number theory

Abstract

In this paper we determine the perfect powers that are sums of three fifth powers in an arithmetic progression. More precisely, we completely solve the Diophantine equation $$ (x-d)^5 + x^5 + (x + d)^5 = z^n,~n\geq 2, $$ where $d,x,z \in \mathbb{Z}$ and $d = 2^a5^b$ with $a,b\geq 0$.