## Number Theory: Lecture 12

In which we finally discover a way to determine which numbers are represented by which forms.  (Well, almost.)

• Lemma 36: Let $\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} p & q \\ r & s \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}$ where $\begin{pmatrix} p & q \\ r & s \end{pmatrix} \in SL_2(\mathbb{Z})$.  Then $x$ and $y$ are coprime if and only if $x'$ and $y'$ are coprime.  This was straightforward to check.
• Definition of what it means for a number to be represented by a form, and what it means for a number to be properly represented by a form.
• 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 $n$.  In one direction, if $f'$ has first coefficient $n$, then it clearly properly represents $n$ (via $(1,0)$).  And if $f$ is equivalent to such a form, then it must also properly represent $n$.  In the other direction, we showed that if $f(p,r) = n$ where $p$ and $r$ are coprime integers, then there is a unimodular substitution under which $(p,r)$ corresponds to $(1,0)$, and then the resulting form is equivalent to $f$ and has first coefficient $n$.
• Theorem 38: Let $n$ be a natural number.
1. Suppose that $n$ is properly represented by a form of discriminant $d$.  Then there is a solution to the congruence $w^2 \equiv d \pmod{4n}$.
2. Suppose that there is a solution to the congruence $w^2 \equiv d \pmod{4n}$.  Then there is a form of discriminant $d$ that properly represents $n$.

For (i), we used Lemma 37, and the fact that if $f$ and $f'$ are equivalent then they have the same discriminant.  For (ii), we took a solution to the congruence, say with $w^2 = d + 4nk$ for some integer $k$, and then immediately obtained a form with discriminant $d = w^2 - 4nk$ that represents $n$, namely $nx^2 + wxy + ky^2$.

We discussed the important fact that in (ii) we cannot in advance be sure which collection of forms will represent $n$.  If there are two or more reduced forms of discriminant $d$, then we get multiple families of numbers represented by these forms.  Nonetheless, the result is still very useful, and in the case when the class number is 1, we get a particularly nice result.

Davenport (The Higher Arithmetic) and Baker (A concise introduction to the theory of numbers) both treat this material.

#### Preparation for Lecture 13

One nice binary quadratic form is the simple $f(x,y) = x^2 + y^2$.  We started discussing this example in the lecture; can you go from the point we reached to a classification of which numbers are the sum of two squares?  You could take this further: which are the sum of three squares?  Or four squares?

### 3 Responses to “Number Theory: Lecture 12”

1. Andrew Says:

Is it OK if we use the Baire Category theorem to answer a question on the example sheet?

2. theoremoftheweek Says:

Now there’s a question I wasn’t expecting!

I think this is between you and your supervisor… I certainly wasn’t expecting anyone to use the Baire category theorem for these problems. Perhaps you could give your supervisor two approaches, one using Baire and one without?

3. Number Theory: Lecture 13 « Theorem of the week Says:

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