On the solutions of $x^2= By^p+Cz^p$ and $2x^2= By^p+Cz^p$ over totally real fields

TL;DR

This paper investigates solutions to specific exponential Diophantine equations over totally real fields, providing new results on existence, non-existence, and local criteria for solutions with prime exponents.

Contribution

It offers novel insights into solutions of equations like $x^2= By^p+Cz^p$ over totally real fields, including primitive solutions and local criteria, extending previous work.

Findings

Characterization of solutions for $x^2= By^p+Cz^p$ over totally real fields.

Identification of conditions for primitive solutions with specific powers of 2.

Development of local criteria for the existence of solutions.

Abstract

In this article, we study the solutions of certain type over $K$ of the Diophantine equation $x^2= By^p+Cz^p$ with prime exponent $p$, where $B$ is an odd integer and $C$ is either an odd integer or $C=2^r$ for $r \in \mathbb{N}$. Further, we study the non-trivial primitive solutions of the Diophantine equation $x^2= By^p+2^rz^p$ ($r\in {1,2,4,5}$) (resp., $2x^2= By^p+2^rz^p$ with $r \in \mathbb{N}$) with prime exponent $p$, over $K$. We also present several purely local criteria of $K$.