## Number Theory: Lecture 12

In which we meet the class number and think about which numbers are represented by which binary quadratic forms.

• Proposition 36: Let $d$ be a fixed negative integer.  There are finitely many reduced forms of discriminant $d$.  We proved this by noting that if $(a,b,c)$ is a reduced form of discriminant $d$ then $d \leq -3a^2$ so there are finitely many possibilities for $a$, and hence for $b$ and $c$.
• Definition of the class number.
• Definition of what it means for a binary quadratic form to represent an integer and to properly represent an integer.
• Lemma 37: Let $f$ be a binary quadratic form.  The integer $n$ is properly represented by $f$ if and only if $f$ is equivalent to a form $f'$ with first coefficient $n$This worked out quite neatly with the help of Bézout.

#### Understanding today’s lecture

Pick a negative integer.  Can you find its class number?  Can you find values of $d$ (that we didn’t study in lectures) with $h(d) = 0$ or $h(d) = 1$ or $h(d) = 2$ or $h(d) \geq 3$?

Pick a positive definite binary quadratic form.  Can you determine (with the help of Lemma 37) which numbers it properly represents?  Is the class number relevant?

As usual, Baker (A concise introduction to the theory numbers) and Davenport (The Higher Arithmetic) have useful things to say, as do these online notes by Andrew Granville.  That Baker is the Alan Baker I mentioned in the lecture, by the way.

I mentioned the class number problem.  There are many places where you can read about this, including this paper by Stark, and this paper by Goldfeld, to give just two examples.  Either of these would give you a sense of the breadth of mathematical ideas that go into the study of the class number.

#### Preparation for Lecture 13

Can you use Lemma 37 to come up with necessary and sufficient conditions for a number to be properly represented by a given form?

Which numbers can be written as sums of two squares?