Binary quadratic forms with the same value set

Abstract

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