On the solutions to $A{x}^p+B{y}^p+C{z}^p=0$ over quadratic fields

Alejandro Argáez-García, Luis Elí Pech-Moreno

We provide the necessary conditions for the existence of solutions $(x,y,z)$ to $A{x}^p+B{y}^p+C{z}^p=0$ over any quadratic number field $K$ with $A,B,C$ pth powerfree integer numbers. W…