Perfect powers that are sums of squares in a three term arithmetic progression

TL;DR

This paper finds primitive solutions to a specific exponential sum of squares in a three-term arithmetic progression, using advanced factorization and primitive divisor theorems, for a range of r values up to 5000.

Contribution

It extends the understanding of sums of squares in arithmetic progressions by explicitly solving for primitive solutions within a large parameter range.

Findings

Identified all primitive solutions for r up to 5000.

Applied factorization and Primitive Divisors Theorem effectively.

Provided new insights into the structure of such exponential sums.

Abstract

We determine primitive solutions to the equation $(x-r)^2 + x^2 + (x+r)^2 = y^n$ for $1 \le r \le 5,000$, making use of a factorization argument and the Primitive Divisors Theorem due to Bilu, Hanrot and Voutier.