On powers that are sums of consecutive like powers.

Let k ≥ 2 be even, and let r be a non-zero integer. We show that for almost all d ≥ 2 (in the sense of natural density), the equation x k + ( x + r ) k + ⋯ + ( x + ( d - 1 ) r ) k = y n , x , y , n ∈ Z , n ≥ 2 , has no solutions.