*In which we develop our understanding of binary quadratic forms.*

- We remarked that while equivalent forms have the same discriminant, the converse is not true: there are forms that have the same discriminant but that are not equivalent. Try to find your own examples, in addition to the one we saw in lectures.
- Lemma 30:
*Let be an integer. Then there is a binary quadratic form with discriminant if and only if is congruent to 0 or 1 modulo 4.*One direction follows from the congruence . To prove the other direction, for each case we simply found a binary quadratic form that does the job. - Definition:
*Positive definite*,*negative definite*, and*indefinite*. - Lemma 31:
*Let be a binary quadratic form with discriminant , and with . If , then is indefinite. If , then is definite. Moreover, if and then is positive definite, and if and then is negative definite.*We proved this by writing . In the case that , the results then follow nicely. We had to do a tiny bit more work to deal with : we noted that the right-hand side is the difference of two squares, and argued from there. - Definition of
*reduced forms*. A positive definite binary quadratic form is*reduced*if or . - Lemma 32:
*Let be a positive definite binary quadratic form. Then is equivalent to a reduced form.*We saw that there were a couple of particularly useful transformations that we could use to reduce forms, namely and . Each of these can be used to reduce one of and while leaving the other fixed, and by doing a suitable combination of these operations we see that we always obtain a form with . Finally, if then we can apply if necessary to ensure that , and if then we can apply to change the middle coefficient to (while leaving the others unchanged).

I gave out the second examples sheet today; you can find it on the course page.

#### Further reading

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

#### Preparation for Lecture 11

We are making good progress with our task of understanding binary quadratic forms. Here are some questions for you to consider before next time.

- For a fixed discriminant , how many reduced forms are there? (Finitely many, or infinitely many?)
- Can you find any discriminants for which there is just one reduced form? Can you find any for which there are exactly two reduced forms?
- Can two reduced forms be equivalent to each other?

October 31, 2011 at 1:37 pm

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