Definition: A Binary Quadratic Form is a polynomial of the form $f(x, y) = ax^2 + bxy + cy^2$ where $a, b, c \in \mathbb{Z}$, not all zero.

For example, $f(x, y) = x^2 + y^2$ is a binary quadratic form with $a = 1$, $b = 0$, and $c = 1$. Another example of a binary quadratic form is $g(x, y) = x^2 - 2xy$ where $a = 1$, $b = 2$, and $c = 0$.

Given a binary quadratic form $f(x, y)$ we want to determine the range of $f$. That is, we want to find all integers $n \in \mathbb{Z}$ such that there exists $x_0, y_0 \in \mathbb{Z}$ with:

(1)
\begin{align} \quad n = f(x_0, y_0) \end{align}
 Definition: Let $f(x, y) = ax^2 + bxy + cy^2$ be a binary quadratic form and let $n \in \mathbb{Z}$. $f(x, y)$ is said to Represent $n$ if there exists $x_0, y_0 \in \mathbb{Z}$ such that $n = f(x_0, y_0)$. Moreover, $f(x, y)$ is said to Properly Represent $n$ if there exists $x_0, y_0 \in \mathbb{Z}$ such that $n = f(x_0, y_0)$ and $(x_0, y_0) = 1$.

For example, let $f(x, y) = x^2 + y^2$ and let $n = 5$. Then:

(2)
\begin{align} \quad 5 = f(1, 2) = 1^2 + 2^2 \end{align}

And $(1, 2) = 1$, so $5$ can be properly represented by $f(x, y)$.