Perfect Powers that are Sums of Squares of an Arithmetic Progression

TL;DR

This paper classifies all primitive solutions where a sum of squares of an arithmetic progression equals a perfect power, for small progression lengths and common differences, using advanced factorization and divisor theorems.

Contribution

It provides a complete characterization of primitive solutions for sums of squares in arithmetic progressions with specified bounds, employing novel application of divisor theorems.

Findings

Identified all primitive solutions for the given parameters.

Applied the Primitive Divisors Theorem to this class of Diophantine equations.

Extended understanding of sums of squares equaling perfect powers in arithmetic progressions.

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 $2\leq d\leq 10$ and for $1\leq r\leq 10^4$. We make use of a factorization argument and the Primitive Divisors Theorem due to Bilu, Hanrot and Voutier.