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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
== Equivalence ==
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"Composition" can also refer to a binary operation on representations of integers by forms. This operation is substantially more complicated{{cn|date=March 2017}}<!--<ref>{{harvnb|Shanks|1989}}</ref> full citation not in article yet --> than composition of forms, but arose first historically. We will consider such operations in a separate section below.
Composition means taking 2 quadratic forms of the same discriminant and combining them to create a quadratic form of the same discriminant, it is a generalization of the 2-square identity <math>\left(a^2 + b^2\right)\left(c^2 + d^2\right) = \left(ac-bd\right)^2 + \left(ad+bc\right)^2</math>
=== Composing forms and classes ===
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==See also==
* [[Bhargava cube]]
*[[Fermat's theorem on sums of two squares]]
* [[Legendre symbol]]
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* Johannes Buchmann, Ulrich Vollmer: ''Binary Quadratic Forms'', Springer, Berlin 2007, {{ISBN|3-540-46367-4}}
* Duncan A. Buell: ''Binary Quadratic Forms'', Springer, New York 1989
* David A Cox, ''Primes of the form <math>x^2 + y^2</math>, Fermat, class field theory, and complex multiplication''
* {{Citation
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