On the power values of the sum of three squares in arithmetic progression

TL;DR

This paper explicitly solves a class of Diophantine equations involving sums of three squares in arithmetic progression, using primitive divisor results, and establishes bounds and uniqueness of solutions under certain conditions.

Contribution

It provides an explicit formula for solutions when n is an odd prime and d is a prime power, improving previous results, and proves solution uniqueness and bounds for general d.

Findings

Explicit solutions for the Diophantine equation when n is an odd prime and d is a prime power.

At most one solution exists under certain conditions.

Solutions are bounded for large n and d.

Abstract

In this paper, using a deep result on the existence of primitive divisors of Lehmer numbers due to Y. Bilu, G. Hanrot and P. M. Voutier, we first give an explicit formula for all positive integer solutions of the Diophantine equation $(x-d)^2+x^2+(x+d)^2=y^n$ (*) when $n$ is an odd prime and $d=p^r$, $p>3$ a prime. So this improves the results on the papers of A. Koutsianas and V. Patel \cite{KP} and A. Koutsianas \cite{Kou}. Secondly, under the assumption of our first result, we prove that (*) has at most one solution $(x,y)$. Next, for a general $d$, we prove the following two results: (i) if every odd prime divisor $q$ of $d$ satisfies $q\not\equiv \pm 1 \pmod{2n},$ then (*) has only the solution $(x,y,d,n)=(21,11,2,3)$. (ii) if $n>228000$ and $d>8\sqrt{2}$, then all solutions $(x,y)$ of (*) satisfy $y^n<2^{3/2}d^3$.