On powers that are sums of consecutive like powers

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 $$x^k+(x+r{)}^k+\cdots +(x+(d-1)r{)}^k=y^n,x,y,n\in \mathbb{Z},n\ge 2,$$ has no solutions.