# Reduced Binary Quadratic Forms

Definition: A positive definite binary quadratic form $f(x, y) = ax^2 + bxy + cy^2$ is said to be Reduced if one of the following inequalities hold:1) $-a < b \leq a < c$.2) $0 < b \leq a = c$. |

Let:

(1)Observe that $\det (M_1) = 1$ and $\det (M_2) = 1$. If $f(x, y) = ax^2 + bxy + cy^2$ then under $M_1$ we have that:

(2)where $a' = c$, $b' = -b$, and $c' = a$. So if $a > c$ then $a' = c < a = c'$. So if $f(x, y)$ is such that the coefficient of $x^2$ is greater than the coefficient of $y^2$ then applying $M_1$ to $f(x, y)$ gives us an equivalent binary quadratic form where the coefficient of $x^2$ is less than the coefficient of $y^2$.

Again, if $f(x, y) = ax^2 + bxy + cy^2$ then under $M_2$ we have that:

(3)where $a' = a$, $b' = 2am + b$, and $c' = am^2 + bm + c$. If $b > a$ then $m$ can be chosen such that:

(4)So if $f(x, y)$ is such that the coefficient of $xy$ is greater than the coefficient of $x^2$ then applying $M_2$ to $f(x, y)$ gives us an equivalent binary quadratic form such that the coefficient of $xy$ is less than the coefficient of $x^2$.

Theorem 1: Let $f(x, y) = ax^2 + bxy + cy^2$ be a reduced binary quadratic form with discriminant $d \in \mathbb{Z}$ where $d$ is not a perfect square.a) If $f$ is indefinite then $0 < |a| \leq \frac{1}{2} \sqrt{d}$.b) If $f$ is positive definite then $0 < a \leq \sqrt{\frac{-d}{3}}$. |

**Proof of a):**Suppose that $f$ is indefinite and is a reduced binary quadratic form. Then $d > 0$ and $-a < b \leq a < c$. If $a$ and $c$ have the same signs then $ac > 0$. So $ac > a^2$. Hence:

- Which is a contradiction. So $a$ and $c$ must have different signs, and hence $-4ac = 4|ac|$. Thus:

- Dividing both sides by $4$ and taking the squareroot of both sides of the inequality gives us:

- If $a = 0$ then the inequality $0 < b \leq a = c$ cannot be satisfied, and so:

**Proof of b)**Suppose that $f$ is positive definite and is a reduced binary quadratic form. Then $d < 0$ and $a, c > 0$, and $-a < b \leq a < c$. So:

- Dividing both sides by $-3$ and taking the squareroot of both sides gives us that:

- The absolute value bars on $a$ are removed since $a > 0$, and so: