On ternary Diophantine equations of signature (p, p, 2) over number fields

Abstract

Let K be a totally real number field with narrow class number one and ${mathcal O}_K$ be its ring of integers. We provethat there is a constant BK depending only on K such that for any prime exponent p > $B_K$ the Fermat type equationmathcal $x^p;+;y^p;=;z^2$ with x, y, z ∈ ${mathcal O}_K$ does not have certain type of solutions. Our main tools in the proof are modularity,level lowering, and image of inertia comparisons.