 Binary quadratic form

A binary quadratic form is a quadratic form in two variables, written $f(x,y)=a{x}^2+bxy+c{y}^2$ or $[a,b,c]$. In number theory an integral binary quadratic form has integer polynomial coefficients $a,b,c$. It represents an integer $n$ when $f(x,y)=n$ for some integers $x,y$.

The homogeneous polynomial identity $f(tx,ty)=t^{2}f(x,y)$ distinguishes a form from a general quadratic polynomial with constant or linear terms. The discriminant of a binary quadratic form is $D=b^2-4ac$. For example, $x^2+y^2$ has $D=-4$ and represents $5=1^2+2^2$, while $x^2+xy+y^2$ has $D=-3$ and represents $7=1+2+4$ at $(1,2)$.

Changing integer coordinates can conceal the same arithmetic. Substituting $(x,y)=(u+v,v)$ into $x^2+y^2$ gives $u^2+2uv+2v^2$. The transformation has determinant one and an integer inverse, so both forms represent exactly the same integers. Proper equivalence of binary quadratic forms records this reversible change, and the reduction algorithm for a positive definite binary quadratic form chooses a small representative for each such equivalence class.