# On perfect powers that are sums of cubes of a five term arithmetic progression

## TL;DR

This paper proves that certain perfect powers expressed as sums of five cubes in an arithmetic progression only have trivial solutions for a large range of parameters, advancing understanding of sums of powers.

## Contribution

It establishes a comprehensive result for the equation involving sums of five cubes in arithmetic progression equaling a perfect power, with solutions only when either x or y is zero.

## Key findings

- Solutions only occur when x or y is zero for the specified range
- No non-trivial solutions exist for p ≥ 5 prime and 1 ≤ r ≤ 10^6
- The result extends the understanding of sums of powers in arithmetic progressions

## Abstract

We prove that the equation $(x-2r)^3 + (x-r)^3 + x^3 + (x+r)^3 + (x+2r)^3= y^p$ only has solutions which satisfy $xy=0$ for $1\leq r\leq 10^6$ and $p\geq 5$ prime.