## Number Theory: Lecture 10

In which we do some more work on understanding binary quadratic forms.

• Lemma 30: Let $d$ be an integer.  There is a binary quadratic form with discriminant $d$ if and only if $d \equiv 0 \pmod{4}$ or $d \equiv 1 \pmod{4}$.  In one direction, we argued that if $(a,b,c)$ has discriminant $d$ then $d \equiv b^2 \pmod{4}$ so it’s a square modulo $4$.  In the other direction, we gave explicit examples of forms with discriminant $d$ if $d \equiv 0 \pmod{4}$ or $d \equiv 1 \pmod{4}$.
• Definition of positive definite, negative definite and indefinite binary quadratic forms.
• Lemma 31: Let $f(x,y) = ax^2 + bxy + cy^2$ be a binary quadratic form with discriminant $d = b^2 - 4ac$ and with $a \neq 0$.  If $d < 0$ and $a > 0$, then $f$ is positive definite.  If $d < 0$ and $a < 0$, then $f$ is negative definite.  If $d > 0$, then $f$ is indefinite.  We proved this by considering $4af(x) = (2ax+by)^2 - dy^2$.
• We defined the substitution $S$ that sends $(a,b,c)$ to $(c,-b,a)$, and the substitutions $T_{\pm}$ that send $(a,b,c)$ to $(a, b\pm 2a, a \pm b + c)$.
• Definition of a reduced positive definite binary quadratic form.

#### Understanding today’s lecture

Pick some binary quadratic forms.  Are they positive definite, negative definite or indefinite?  Which of the positive definite forms are reduced?  Can you use the substitutions $S$ and $T_{\pm}$ to reduce the non-reduced positive definite forms?