In which we finally discover a way to determine which numbers are represented by which forms. (Well, almost.)
- Lemma 36: Let where . Then and are coprime if and only if and 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 be a binary quadratic form. The integer is properly represented by if and only if is equivalent to a form with first coefficient . In one direction, if has first coefficient , then it clearly properly represents (via ). And if is equivalent to such a form, then it must also properly represent . In the other direction, we showed that if where and are coprime integers, then there is a unimodular substitution under which corresponds to , and then the resulting form is equivalent to and has first coefficient .
- Theorem 38: Let be a natural number.
- Suppose that is properly represented by a form of discriminant . Then there is a solution to the congruence .
- Suppose that there is a solution to the congruence . Then there is a form of discriminant that properly represents .
For (i), we used Lemma 37, and the fact that if and are equivalent then they have the same discriminant. For (ii), we took a solution to the congruence, say with for some integer , and then immediately obtained a form with discriminant that represents , namely .
We discussed the important fact that in (ii) we cannot in advance be sure which collection of forms will represent . If there are two or more reduced forms of discriminant , 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.
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 . 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?