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.

Further reading

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?

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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:

    […] Theorem of the week Expositions of interesting mathematical results « Number Theory: Lecture 12 […]

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