integral binary quadratic forms

An integral binary quadratic form is a quadratic formMathworldPlanetmath (q.v.) in two variables over , i.e. a polynomial


F is said to be primitivePlanetmathPlanetmath if its coefficients are relatively prime, i.e. gcd(a,b,c)=1, and is said to represent an integer n if there are r,s such that F(r,s)=n. If gcd(r,s)=1, F is said to represent n properly. The theory of integral binary quadratic forms was developed by Gauss, Lagrange, and Legendre.

In what follows, “form” means “integral binary quadratic form”.

Following the article on quadratic forms, two such forms F(x,y) and G(x,y) are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath if there is a matrix MGL(2,) such that


Matrices in GL(2,) are matrices with determinant ±1. So if α,β,γ,δ and


then if


it follows that G is equivalent to F. If MSL(2,) (i.e. detM=1), we say that F and G are properly equivalent, written FG; otherwise, they are improperly equivalent.

Note that while both equivalence and proper equivalence are equivalence relations, improper equivalence is not. For if F is improperly equivalent to G and G is improperly equivalent to H, then the productPlanetmathPlanetmath of the transformation matrices has determinant 1, so that F is properly equivalent to H. Since proper equivalence is an equivalence relation, we will say that two forms are in the same class if they are properly equivalent.

GL(2,) is generated as a multiplicative groupMathworldPlanetmath by the two matrices


so in particular we see that we can construct all equivalence transformationsMathworldPlanetmath by composing the following three transformations:

Transformation Matrix Determinant
(x,y)(y,x) (0110) -1
(x,y)(y,-x) (0-110) 1
(x,y)(x+dy,y) (10d1) 1

Example: Let F(x,y)=x2+xy+6y2, G(x,y)=82x2+51xy+8y2. Then




The transformations to map F into G are



Title integral binary quadratic forms
Canonical name IntegralBinaryQuadraticForms
Date of creation 2013-03-22 16:55:44
Last modified on 2013-03-22 16:55:44
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 10
Author rm50 (10146)
Entry type Topic
Classification msc 11E16
Classification msc 11E12
Related topic RepresentationOfIntegersByEquivalentIntegralBinaryQuadraticForms
Related topic ReducedIntegralBinaryQuadraticForms