Perfect powers that are sums of squares of an arithmetic progression

Abstract

We determine all nontrivial integer solutions to the equation ${\left(x+r\right)}^2+{\left(x+2r\right)}^2+\cdots +{\left(x+dr\right)}^2=y^n$ for $2\le d\le 10$ and $1\le r\le 10^4$ with $gcd\left(x,y\right)=1$. We make use of a factorization argument and the primitive divisors theorem due to Bilu, Hanrot and Voutier.