representation of integers by equivalent integral binary quadratic forms
properly represents an integer if and only if is properly equivalent to a form .
: It is obvious by the above that represents ; the problem is to show that it represents properly. Write ; then , where . Then . But clearly since otherwise we cannot have . So represents properly.
: Write , where . Since , we can find integers such that , and then
Note that is always either congruent to or mod 4, and that is even (odd) exactly when .
If are equivalent integral quadratic forms, then .
For any form , define
Note further that .
Now in our particular case, if , then
But , so since ,
Example: In the previous example, note that , and .
The converse of this theorem is not true - that is, there are forms of the same discriminant that represent different numbers. For example, and both have discriminant , yet the second form represents while the first clearly does not. However, equivalence classes of forms under arbitrary (proper or improper) equivalence represent disjoint sets of primes:
Let be an odd prime. Suppose both represent and . Then and are equivalent (but perhaps not properly equivalent).
Since is prime, obviously represents properly. So . Note that the transformation results in a form whose middle term is , so by an appropriate choice of we can arrange that . Similarly, with . Note also that since , it follows that (i.e. have the same parity).
Since , we see that , so for some . Since have the same parity and is odd, is even; since , (since otherwise would be separated by at least , which is impossible).
We are left with two cases. If , then implies that and hence . If , then again implies that . Then and are equivalent via the transformation . ∎
Note that and are always improperly equivalent via the transformation . They are sometimes properly equivalent, and sometimes not. For example, and are properly equivalent while and are not. (See the article on reduced integral binary quadratic forms for details).
In summary, we have proved the following:
We conclude with the following lemma and corollary, which provide concrete criteria for when an integer is representable by a class of forms.
If properly represents , then by the preceding lemma, we may assume that . Then , being the discriminant of , so that . Conversely, if , we may assume (if they have different parities, replace by ; since is odd, the condition now holds and as well). Since , it follows that and thus . Hence for some integer . But then represents and has discriminant ; it is primitive since . ∎
Let be an integer, and an odd prime not dividing . Then if and only if is represented by a primitive form of discriminant .
By the preceding lemma, is represented by a primitive form of discriminant if and only if
|Title||representation of integers by equivalent integral binary quadratic forms|
|Date of creation||2013-03-22 19:18:48|
|Last modified on||2013-03-22 19:18:48|
|Last modified by||rm50 (10146)|