On the solutions of $x^p+y^p=2^r z^p$, $x^p+y^p=z^2$ over totally real fields

TL;DR

This paper investigates solutions to specific exponential Diophantine equations over totally real fields, providing conditions under which no non-trivial solutions exist, especially for certain exponents and local criteria.

Contribution

It offers new results on the existence of solutions to these equations over totally real fields, including criteria that guarantee no solutions for particular cases.

Findings

No non-trivial solutions for certain exponents over totally real fields.

Local criteria that prevent solutions over the ring of integers.

Analysis of solutions for specific cases r=2,3.

Abstract

In this article, we study the non-trivial primitive solutions of a certain type for the Diophantine equations $x^p+y^p=2^rz^p$ and $x^p+y^p=z^2$ of prime exponent $p$, $r \in \mathbb{N}$, over a totally real field $K$. Then for $r=2,3$, we study the non-trivial primitive solutions over $\mathcal{O}_K$ for the equation $x^p+y^p=2^rz^p$ of prime exponent $p$. Finally, we give several purely local criteria for $K$ such that the equation $x^p+y^p=2^rz^p$ has no non-trivial primitive solutions over $\mathcal{O}_K$.