there is a unique reduced form of discriminant only for
The number of reduced http://planetmath.org/node/IntegralBinaryQuadraticFormsintegral binary quadratic forms of a given http://planetmath.org/node/IntegralBinaryQuadraticFormsdiscriminant is finite; the number of such forms is .
Let be a positive integer. Then if and only if .
(This proof is taken from , which is itself taken from an earlier proof).
By computing all reduced forms of discriminants one can see that in the five cases given in the statement of the theorem. We show there are no others.
Clearly is a reduced form of discriminant . For , we will produce a second reduced form of the same discriminant, showing that . We may assume , since we already know that .
Suppose first that has at least two distinct prime factors. Then we can write where . Then is reduced, and its discriminant is . So if has two distinct prime factors, .
We now consider the prime power case, taking and , an odd prime, separately.
If , then we already know that for , . For , one can compute the classes of discriminant and see that . For , then
is clearly primitive, and is also reduced since . Further, its discriminant is . Thus in this case as well, .
Finally, suppose , an odd prime. Suppose we can write . Then
is reduced and has discriminant , so . So we are left with the case where is a prime power which, since it is even, must be . correspond to ; corresponds to , which is not a prime power; and for , one can simply compute the forms of discriminant to see that . So the only possibility remaining is that . In this case, though,
has relatively prime coefficients, and is reduced since . Also, its discriminant is , and thus in this case as well. ∎
|Title||there is a unique reduced form of discriminant only for|
|Date of creation||2013-03-22 16:56:46|
|Last modified on||2013-03-22 16:56:46|
|Last modified by||rm50 (10146)|