reduced integral binary quadratic forms

Definition 1.

A positive binary form is one where F(x,y)0x,yZ.

This article deals only with positive integral binary quadratic forms (i.e. those with negative discriminantPlanetmathPlanetmath and with a>0). Some but not all of this theory applies to forms with positive discriminant.

Proposition 1.

If F is positive, then a>0. If Δ(F)<0 and either a>0 or c>0, then F is positive.


If F is positive, then F(1,0)=a, so a>0.
If Δ(F)<0, then 4aF(x,y)=(2ax+by)2-Δy20. Thus if a>0, then F(x,y)0. The proof for the case c>0 is identical. ∎

Definition 2.

A primitive positive form ax2+bxy+cy2 is reduced if

|b|ac, and b0 if either |b|=a or a=c

This is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to saying that -abac and that b can be negative only if a<c or -a<b. Thus (3,-2,4) is reduced, but (3,-2,3) is not.

It turns out that each proper equivalence classMathworldPlanetmath of primitive positive forms contains a single reduced form, and thus we can understand how many classes there are of a given discriminant by studying only the reduced forms.

Theorem 2.

If F is a primitive positive reduced form with discriminant Δ-3, F(x,y)=ax2+bxy+cy2, then the minimum value assumed by F if x,y are not both zero is a. If a<c, then this value is assumed only for (x,y)=(±1,0); if a=c; it is assumed for (x,y)=(±1,0) and (x,y)=(0,±1).


Since |b|ac, it follows that


Thus, F(x,y)a-|b|+c whenever xy0, while if x or y is zero, then F(x,y)a. So a is clearly the smallest nonzero value of F.

If a<c, then F(x,y)a+(c-|b|)>a if xy0, and F(0,y)c>a for y0, so F achieves its minimum only at (x,y)=(±1,0).

If a=c, then |b|a since otherwise F is not reduced (we cannot have a=c=1,b=±1 else we have a form of discriminant -3). Thus again c-|b|>0 and thus F(x,y)>a if xy0, so in this case the result follows as well. ∎

Note that the reduced form of discriminant -3, x2+xy+y2, also achieves its minimum value at (1,-1),(-1,1).

Theorem 3.

If F,G are primitive positive reduced forms with FG, then F=G.


We take the cases Δ-3 and Δ=-3 separately.

First assume Δ-3 so we can apply the above theorem.

Since FG, we can write G(x,y)=F(αx+βy,γx+δy) with αδ-βγ=1. Suppose F=ax2+bxy+cy2,G=ax2+bxy+cy2. Now, F and G have the same minimum value, so a=a.

If a<c, then F achieves its minimum only at (pm1,0), so a=a=G(1,0)=F(α,γ) and thus α=±1,γ=0. So G(x,y)=F(±x+ry,±y) and thus b=b+2ra. Since G is also reduced, b=b and thus c=c and F=G.

If instead a=c, then instead of concluding that α=±1 we can only conclude that α=±1 or γ=±1. If α=±1, the argument carries through as above. If γ=±1, then α=0,β=1, so G(x,y)=F(y,±x+ry) and thus b=±2cr-b. Thus b=-b. But then c=c since the discriminants are equal, and thus both b,b0. So b=b=0 and we are done.

Finally, in the case Δ=-3, we see that for any such reduced form, 3=4ac-b24a2-a2=3a2, so a=1,b=±1,c=1. Thus b=1 since otherwise the form is not reduced. So the only reduced form of discriminant -3 is in fact x2+xy+y2. ∎

Theorem 4.

Every primitive positive form is properly equivalent to a unique reduced form.


We just proved uniqueness, so we must show existence. Note that I used a different method of proof in class that relied on “infinite descent” to get the result in the first paragraph below; the method here is just as good but provides less insight into how to actually reduce a form.

We first show that any such form is properly equivalent to some form satisfying |b|ac. Among all forms properly equivalent to the given one, choose F(x,y)=ax2+bxy+cy2 such that |b| is as small as possible (there may be multipleMathworldPlanetmathPlanetmath such forms; choose one of them). If |b|>a, then


is properly equivalent to F, and we can choose m so that |2am+b|<|b|, contradicting our choice of minimal |b|. So |b|a; similarly, |b|c. Finally, if a>c, simply interchange a and c (by applying the proper equivalence (x,y)(-y,x)) to get the required form.

To finish the proof, we show that if F(x,y)=ax2+bxy+cy2, where |b|ac, then F is properly equivalent to a reduced form. The form is already reduced unless b<0 and either a=-b or a=c. But in these cases, the form G(x,y)=ax2-bxy+cy2 is reduced, so it suffices to show that F and G are properly equivalent. If a=-b, then (x,y)(x+y,y) takes ax2-axy+cy2 to ax2+axy+cy2, while if a=c, then (x,y)(-y,x) takes ax2+bxy+ay2 to ax2-bxy+ay2. ∎

Let’s see how to reduce 82x2+51xy+8y2 to x2+xy+6y2:

Form TransformationPlanetmathPlanetmath Result
82x2+51xy+8y2 (x,y)(-y,x) 8x2-51xy+82y2
8x2-51xy+82y2 (x,y)(x+3y,y) 8(x+3y)2-51(x+3y)y+82y2=
8x2-3xy+y2 (x,y)(-y,x) x2+3xy+8y2
x2+3xy+8y2 (x,y)(x-y,y) (x-y)2+3(x-y)y+8y2=x2+xy+6y2
Theorem 5.

If F(x,y)=ax2+bxy+cy2 is a positive reduced form with Δ<0, then a|Δ|3.


-Δ=4ac-b24a2-a2 since the form is reduced. So -Δ3a2, and the result follows. ∎

Definition 3.

If Δ<0, then h(Δ) is the number of classes of primitive positive forms of discriminant Δ.

Corollary 6.

If Δ<0, then h(Δ) is finite, and h(Δ) is equal to the number of primitive positive reduced forms of discriminant Δ.


Essentially obvious. Since every positive form is properly equivalent to a (unique) reduced form, h(Δ) is clearly equal to the number of positive reduced forms of discriminant Δ. But given a reduced form of discriminant Δ, there are only finitely many choices for a>0, by the propositionPlanetmathPlanetmath. This constrains us to finitely many choices for b, since -a<ba. a and b determine c since Δ is fixed. ∎

Examples: Δ=-4: b2-4ac=-4b even, |b||a|43b=0. So (1,0,1), corresponding to x2+y2, is the only reduced form of discriminant -4. Note that this provides another proof that primes 1(4) are representable as the sum of two squares, since all such primes have (-4p)=(-1p)=1 and thus are representable by this quadratic formMathworldPlanetmath.

Δ=-23: b2-4ac=-23b odd, |b|a233b=±1. So ac=6,a<c. This gives us

(1,-1,6) not reduced since |b|=a,b<0; properly equivalent to (1,1,6) via (x,y)(x+y,y)
(2,-1,3) reduced since b|a,ac

There are three equivalence classes of positive reduced forms with discriminant -23.

Δ=-55: 4ac-b2=55, so b is odd, |b|553|b|=1,±3. So ac=14 or 16,ac. So the forms are

(1,-1,14) not reduced since |b|=a,b<0, properly equivalent to (1,1,14) via (x,y)(x+y,y)
(2,-1,7) reduced since |b|a,ac
(4,-3,4) not reduced since a=c,b<0, equivalent to (4,3,4) via (x,y)(-y,x)

There are four classes of forms of discriminant -55.

Δ=-163: b2-4ac=-163b odd, |b||a|163355. So b=±1,±3,±5,±7, and ac=b2+1634, so ac=41,43,45,47. Since a<c, we must have a=1; thus b=±1 and thus we get only (1,±1,41). But (1,-1,41) is properly equivalent to (1,1,41) via (x,y)(x+y,y), so there is only one equivalence class of positive reduced forms with discriminant -163.

Title reduced integral binary quadratic forms
Canonical name ReducedIntegralBinaryQuadraticForms
Date of creation 2013-03-22 19:18:52
Last modified on 2013-03-22 19:18:52
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 4
Author rm50 (10146)
Entry type Definition
Classification msc 11E12
Classification msc 11E16
Related topic integralbinaryquadraticforms