there is a unique reduced form of discriminant -4n only for n=1,2,3,4,7

The number of reduced binary quadratic forms of a given Δ<0 is finite; the number of such forms is hΔ.

Theorem 1.

Let n be a positive integer. Then h-4n=1 if and only if n=1,2,3,4, or 7.


(This proof is taken from [1], which is itself taken from an earlier proof).
By computing all reduced forms of discriminants -4,-8,-12,-16,-20 one can see that h-4n=1 in the five cases given in the statement of the theorem. We show there are no others.

Clearly x2+ny2 is a reduced form of discriminant -4n. For n{1,2,3,4,7}, we will produce a second reduced form of the same discriminant, showing that h-4n>1. We may assume n>1, since we already know that h-4=1.

Suppose first that n has at least two distinct prime factors. Then we can write n=ac where 1<a<c,gcd(a,c)=1. Then ax2+cy2 is reduced, and its discriminant is -4ac=-4n. So if n has two distinct prime factors, h-4n>1.

We now consider the prime power case, taking 2r and pr, p an odd prime, separately.

If n=2r, then we already know that for r=1,2, h-4n=1. For r=3, one can compute the classes of discriminant -32 and see that h-4n=2. For r4, then


is clearly primitive, and is also reduced since 42r-2+1. Further, its discriminant is 42-44(2r-2+1)=-162r-2=-4n. Thus in this case as well, h-4n>1.

Finally, suppose n=pr, p an odd prime. Suppose we can write n+1=ac,2a<c,gcd(a,c)=1. Then


is reduced and has discriminant -4n, so h-4n>1. So we are left with the case where n+1 is a prime power which, since it is even, must be 2s. s=1,2,3 correspond to n=1,3,7; s=4 corresponds to n=15, which is not a prime power; and for s=5, one can simply compute the forms of discriminant -431 to see that h-431=3. So the only possibility remaining is that s6. In this case, though,


has relatively prime coefficients, and is reduced since s682s-3+1. Also, its discriminant is 62-48(2s-3+1)=4-42s=4-4(n+1)=-4n, and thus h-4n>1 in this case as well. ∎


Title there is a unique reduced form of discriminant -4n only for n=1,2,3,4,7
Canonical name ThereIsAUniqueReducedFormOfDiscriminant4nOnlyForN12347
Date of creation 2013-03-22 16:56:46
Last modified on 2013-03-22 16:56:46
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 6
Author rm50 (10146)
Entry type Theorem
Classification msc 11E12
Classification msc 11R29
Classification msc 11E16