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

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