  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 $\mathcal{O}_K$.
