Integers of a quadratic field with prescribed sum and product

Abstract

For given $k,\ell \in \mathbb Z$ we study the Diophantine system $$x+y+z=k, \quad x y z = \ell $$ for $x,y,z$ integers in a quadratic number field, which has a history in the literature. When $\ell =1$, we describe all such solutions; only for $k=5,6$, do