Binary quadratic form

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In mathematics, a binary quadratic form is a quadratic homogeneous polynomial in two variables

${\displaystyle q(x,y)=ax^{2}+bxy+cy^{2},\,}$

where a, b, c are the coefficients. When the coefficients can be arbitrary complex numbers, most results are not specific to the case of two variables, so they are described in quadratic form. A quadratic form with integer coefficients is called an integral binary quadratic form, often abbreviated to binary quadratic form.

This article is entirely devoted to integral binary quadratic forms. This choice is motivated by their status as the driving force behind the development of algebraic number theory. Since the late nineteenth century, binary quadratic forms have given up their preeminence in algebraic number theory to quadratic and more general number fields, but advances specific to binary quadratic forms still occur on occasion.

Pierre Fermat stated that if p is an odd prime then the equation ${\displaystyle p=x^{2}+y^{2}}$ has a solution iff ${\displaystyle p\equiv 1{\pmod {4}}}$, and he made similar statement about the equations ${\displaystyle p=x^{2}+2y^{2}}$, ${\displaystyle p=x^{2}+3y^{2}}$, ${\displaystyle p=x^{2}-2y^{2}}$ and ${\displaystyle p=x^{2}-3y^{2}}$. ${\displaystyle x^{2}+y^{2},x^{2}+2y^{2},x^{2}-3y^{2}}$ and so on are quadratic forms, and the theory of quadratic forms gives a unified way of looking at and proving these theorems.

Another instance of quadratic forms is Pell's equation ${\displaystyle x^{2}-ny^{2}=1}$.

Binary quadratic forms are closely related to ideals in quadratic fields. This allows the class number of a quadratic field to be calculated by counting the number of reduced binary quadratic forms of a given discriminant.

The classical theta function of 2 variables is ${\displaystyle \sum _{(m,n)\in \mathbb {Z} ^{2}}q^{m^{2}+n^{2}}}$, if ${\displaystyle f(x,y)}$ is a positive definite quadratic form then ${\displaystyle \sum _{(m,n)\in \mathbb {Z} ^{2}}q^{f(m,n)}}$ is a theta function.