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 $F(\mathbb{Z}^2) = G(\mathbb{Z}^2)$.