In this article, we study the solutions of certain type over K of the
Diophantine equation x2=Byp+Czp with prime exponent p, where B is an
odd integer and C is either an odd integer or C=2r for rβN.
Further, we study the non-trivial primitive solutions of the Diophantine
equation x2=Byp+2rzp (rβ1,2,4,5) (resp., 2x2=Byp+2rzp with
rβN) with prime exponent p, over K. We also present several
purely local criteria of K.Comment: Submitted for publication; Any comments are welcome. arXiv admin
note: text overlap with arXiv:2207.1093