On powers that are sums of consecutive like powers

TL;DR

This paper investigates the rarity of sums of consecutive like powers equaling perfect powers, proving that for most parameters, such equations have no solutions, thus extending understanding of power sum equations.

Contribution

It establishes that for almost all relevant parameters, the sum of consecutive like powers cannot be a perfect power, providing new density-based non-existence results.

Findings

Almost all sums of consecutive like powers are not perfect powers.

The result applies for even exponents and almost all step sizes.

The paper advances the understanding of power sum equations in number theory.

Abstract

Let $k \ge 2$ be even, and let $r$ be a non-zero integer. We show that for almost all $d \ge 2$ (in the sense of natural density), the equation $$ x^k+(x+r)^k+\cdots+(x+(d-1)r)^k=y^n, \qquad x,~y,~n \in \mathbb{Z}, \qquad n \ge 2, $$ has no solutions.