Nontrivial solutions to the Fermat equation $x^3+y^3=kz^3$ over quadratic number fields

Abstract

We give sufficient conditions to determine the existence of nontrivial solutions to the Fermat equation $x^3+y^3=kz^3$ over $\mathbb{Q}(\sqrt{d})$ by constructing a relationship with the points on the elliptic curve $y^2=x^3-432d^3k^2$ over $\mathbb{Q}$ for certain $k\in\mathbb{N}$.