integral binary quadratic forms
is said to be primitive if its coefficients are relatively prime, i.e. , and is said to represent an integer if there are such that . If , is said to represent 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 and are equivalent if there is a matrix such that
Matrices in are matrices with determinant . So if and
it follows that is equivalent to . If (i.e. ), we say that and are properly equivalent, written ; otherwise, they are improperly equivalent.
Note that while both equivalence and proper equivalence are equivalence relations, improper equivalence is not. For if is improperly equivalent to and is improperly equivalent to , then the product of the transformation matrices has determinant , so that is properly equivalent to . Since proper equivalence is an equivalence relation, we will say that two forms are in the same class if they are properly equivalent.
is generated as a multiplicative group by the two matrices
so in particular we see that we can construct all equivalence transformations by composing the following three transformations:
Example: Let , . Then
The transformations to map into are
|Title||integral binary quadratic forms|
|Date of creation||2013-03-22 16:55:44|
|Last modified on||2013-03-22 16:55:44|
|Last modified by||rm50 (10146)|