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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 field|quadratic]] and more general [[number field]]s, but advances specific to binary quadratic forms still occur on occasion.
Pierre Fermat stated that if p is an odd prime then the equation <math>p = x^2 + y^2</math> has a solution iff <math>p \equiv 1 \pmod{4}</math>, and he made similar statement about the equations <math>p = x^2 + 2y^2</math>, <math>p = x^2 + 3y^2</math>, <math>p = x^2 - 2y^2</math> and <math>p = x^2 - 3y^2</math>.
<math>x^2 + y^2, x^2 + 2y^2, x^2 - 3y^2</math> 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]] <math>x^2-ny^2=1</math>.
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 <math> \sum_{(m,n)\in \mathbb{Z}^2} q^{m^2 + n^2}</math>, if <math>f(x,y)</math> is a positive definite quadratic form then <math> \sum_{(m,n)\in \mathbb{Z}^2} q^{f(m,n)}</math> is a theta function.
== Equivalence ==
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