Perfect Powers that are Sums of Squares of an Arithmetic Progression

Abstract

In this paper, we determine all primitive solutions to the equation $(x+r{)}^2+(x+2r{)}^2+\cdots +(x+dr{)}^2=y^n$ for $2\le d\le 10$ and for $1\le r\le 10^4$. We make use of a factorization argument and the Primitive Divisors Theorem due to Bilu, Hanrot and Voutier.

18 pages, 3 tables In this version, we substantially improve Theorem 1.3 from the previous version in order to solve the case $d=6$ and $1\le r\le 10^4$. We are now able to record Theorems 1.1, 1.2 and 1.3 from the previous version in one concise theorem