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?

Further reading

Davenport (The Higher Arithmetic) discusses this material, as does Baker (A concise introduction to the theory of numbers).

Preparation for Lecture 11

Is every positive definite binary quadratic form equivalent to a reduced form?

Can two reduced forms be equivalent?

How many reduced forms are there with a given discriminant?  (You could try some particular discriminants and do a direct calculation.)


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