In which we explore reduced binary quadratic forms.
- Lemma 32: Let be a positive definite binary quadratic form. Then is equivalent to a reduced form. We used three unimodular substitutions ( and ) and gave an algorithm for reducing a form.
- Lemma 33: Let be a reduced form. The smallest integers represented by for coprime and (or ) are , , and , in that order. We noted that and and , and then checked what happens if or .
- Lemma 34: Let where . The integers and are coprime if and only if and are coprime. This was a straightforward check.
- Theorem 35: Each positive definite binary quadratic form is equivalent to a unique reduced form. We used Lemma 32, and then showed that if two reduced forms are equivalent then they are the same, using Lemmas 33 and 34.
Understanding today’s lecture
Pick some positive definite binary quadratic forms and reduce them.
Baker (A concise introduction to the theory numbers) covers all of this material. Davenport (The Higher Arithmetic) doesn’t go into the details of the proof of Theorem 35, but discusses many of the other ideas in today’s lecture. There are some online notes by Andrew Granville that cover this material.
Preparation for Lecture 12
How many reduced forms are there with a given discriminant? (E.g. are there infinitely many or finitely many, or does it depend on the discriminant?)
Next time we’ll be tackling the question of which numbers can be represented by binary quadratic forms.
If is a binary quadratic form, then can certainly be represented (because ). It would be interesting to know whether the converse is true, at least once we’re clear what the converse might be. Let be a form that represents . We know that might not have first coefficient , not least because there are many forms equivalent to that represent the same set of numbers, and they don’t all have the same first coefficient. But is it the case that if represents then is equivalent to a form with first coefficient ?