On the solutions of the Diophantine equation $(x-d)^2+x^2+(x+d)^2=y^n$ for $d$ a prime power

TL;DR

This paper classifies primitive solutions to a specific Diophantine equation involving prime power parameters, using advanced divisor theory and algorithmic methods to prove the non-existence of solutions in many cases.

Contribution

It introduces a novel algorithmic approach combined with Lehmer sequence divisor analysis to solve a class of Diophantine equations with prime power parameters.

Findings

Primitive solutions are fully characterized for certain prime power parameters.

The developed algorithm effectively proves the non-existence of solutions in many cases.

The approach advances methods for solving exponential Diophantine equations.

Abstract

In this paper, we determine the primitive solutions of the Diophantine equation $(x-d)^2+x^2+(x+d)^2=y^n$ when $n\geq 2$ and $d=p^b$, $p$ a prime and $p\leq 10^4$. The main ingredients are the characterization of primitive divisors on Lehmer sequences and the development of an algorithmic method of proving the non-existence of integer solutions of the equation $f(x)=a^b$, where $f(x)\in\mathbb Z[x]$, $a$ a positive integer and $b$ an arbitrary positive integer.