In which we do some more work on understanding binary quadratic forms.
- Lemma 30: Let be an integer. There is a binary quadratic form with discriminant if and only if or . In one direction, we argued that if has discriminant then so it’s a square modulo . In the other direction, we gave explicit examples of forms with discriminant if or .
- Definition of positive definite, negative definite and indefinite binary quadratic forms.
- Lemma 31: Let be a binary quadratic form with discriminant and with . If and , then is positive definite. If and , then is negative definite. If , then is indefinite. We proved this by considering .
- We defined the substitution that sends to , and the substitutions that send to .
- 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 and to reduce the non-reduced positive definite forms?
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.)