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

Abstract

In this paper, we determine the primitive solutions of the Diophantine equation $(x-d{)}^2+x^2+(x+d{)}^2=y^n$ when $n\ge 2$ and $d=p^b$, $p$ a prime and $p\le 10^4$. 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.

arXiv commentWe have completely changed the methodology compare to the first version. For fix $d$ the new approach has already been used by Vandita Patel and the author in an earlier paper https://doi.org/10.1142/S1793042118501646 and we extend it when $d$ is an arbitrary prime power