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

In this article, we study the solutions of certain type over a totally real number field 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