Binary quadratic forms with the same value set

Abstract

Given a binary quadratic form $F\in \mathbb{Z}[X,Y]$, we define its value set $F({\mathbb{Z}}^2)$ to be $\{F(x,y):(x,y)\in {\mathbb{Z}}^2\}$. If $F$ and $G$ are two binary quadratic forms with integer coefficients, we give necessary and sufficient conditions on $F$ and $G$ for $F({\mathbb{Z}}^2)=G({\mathbb{Z}}^2)$.