Solving equations of signature $(p,p,2)$ with coefficients over number fields

Abstract

Using the modular method, we study solutions to the Diophantine equation $$A{a}^p+B{b}^p=C{c}^2$$ over number fields. We first prove an asymptotic result for general number fields satisfying an appropriate $S$-unit condition by assuming some standard conjectures in the case of fields that are not totally real. Specifically, we verify that this condition holds for an infinite family of real quadratic fields. Outside the asymptotic setting, we also obtain effective results. In particular, for the equation $$a^p+d{b}^p=c^2$$ over $K=\mathbb{Q}(\sqrt{-d})$ with $d\in \{3,11,19,43,\}$ and $K=\mathbb{Q}(\sqrt{d})$ with $d\in \{3,5,11,13,19,29\}$, we find explicit bounds (depending on $d$) such that no non-trivial solutions of a certain type exist whenever $p$ exceeds these bounds.