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 \(x^2= By^p+Cz^p\) $x^2=B{y}^p+C{z}^p$ with prime exponent p, where B is an odd integer and C is either an odd integer or \(C=2^r\) $C=2^r$ for \(r \in \mathbb {N}\) $r\in \mathbb{N}$ . Further, we study the non-trivial primitive solutions of the Diophantine equation \(x^2= By^p+2^rz^p\) $x^2=B{y}^p+2^r{z}^p$ ( \(r\in {1,2,4,5}\) $r\in 1,2,4,5$ ) (resp., \(2x^2= By^p+2^rz^p\) $2x^2=B{y}^p+2^r{z}^p$ with \(r \in \mathbb {N}\) $r\in \mathbb{N}$ ) with prime exponent p, over K. We also present several purely local criteria of K