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 solutions of the Diophantine equations $x^p+y^p=2^r z^p$ and $x^p+y^p=z^2$ of prime exponent $p$ with $r \in \mathbb{N}$ over a totally real field $K$. We show that there exists a constant $B_{K,r}>0$, depending on $K,r$, such that for all primes $p>B_{K,r}$, the equation $x^p+y^p=2^rz^p$ (resp., $x^p+y^p=z^2$) of exponent $p$ has no non-trivial solution in $W_K$(resp., in $W_K^\prime$). Then, for $r=2,3$, we show that there exists a constant $B_K>0$ such that for all primes $p>B_K$, the equation $x^p+y^p=2^r z^p$ of exponent $p$ has no non-trivial solution over $K$