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The examples above discuss the representation problem for the numbers 3 and 65 by the form <math>x^2 + y^2</math> and for the number 1 by the form <math>x^2 - 2y^2</math>. We see that 65 is represented by <math>x^2 + y^2</math> in sixteen different ways, while 1 is represented by <math>x^2 - 2y^2</math> in infinitely many ways and
3 is not represented by <math>x^2+y^2</math> at all. In the first case, the sixteen representations were explicitly described. It was also shown that the number of representations of an integer by <math>x^2+y^2</math> is always finite. The [[sum of squares function]] <math>r_2(n)</math> gives the number of representations of ''n'' by <math>x^2+y^2</math> as a function of ''n''. There is a closed formula<ref>{{harvnb|Hardy|Wright|2008
: <math> r_2(n) = 4(d_1(n) - d_3(n)), </math>
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has determinant 1 and is an automorphism of ''f''. Acting on the representation <math>1 = f(x_1,y_1)</math> by this matrix yields the equivalent representation <math>1 = f(3x_1 + 4y_1, 2x_1 + 3 y_1)</math>. This is the recursion step in the process described above for generating infinitely many solutions to <math>1 = x^2 - 2y^2</math>. Iterating this matrix action, we find that the infinite set of representations of 1 by ''f'' that were determined above are all equivalent.
There are generally finitely many equivalence classes of representations of an integer ''n'' by forms of given nonzero discriminant <math>\Delta</math>. A complete set of representatives for these classes can be given in terms of ''reduced forms'' defined in the section below. When <math>\Delta < 0</math>, every representation is equivalent to a unique representation by a reduced form, so a complete set of representatives is given by the finitely many representations of ''n'' by reduced forms of discriminant <math>\Delta</math>. When <math>\Delta > 0</math>, Zagier proved that every representation of a positive integer ''n'' by a form of discriminant <math>\Delta</math> is equivalent to a unique representation <math>n = f(x,y)</math> in which ''f'' is reduced in Zagier's sense and <math>x > 0</math>, <math>y \geq 0</math>.<ref>{{harvnb|Zagier|1981
== Reduction and class numbers<!--'Class number (binary quadratic forms)' redirects here--> ==
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Lagrange proved that for every value ''D'', there are only finitely many classes of binary quadratic forms with discriminant ''D''. Their number is the '''{{vanchor|class number}}'''<!--boldface per WP:R#PLA--> of discriminant ''D''. He described an algorithm, called '''reduction''', for constructing a canonical representative in each class, the '''reduced form''', whose coefficients are the smallest in a suitable sense.
Gauss gave a superior reduction algorithm in ''[[Disquisitiones Arithmeticae]]'', which ever since has been the reduction algorithm most commonly given in textbooks. In 1981, Zagier published an alternative reduction algorithm which has found several uses as an alternative to Gauss's.<ref>{{harvnb|Zagier|1981
== Composition ==
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"Composition" also sometimes refers to, roughly, a binary operation on binary quadratic forms. The word "roughly" indicates two caveats: only certain pairs of binary quadratic forms can be composed, and the resulting form is not well-defined (although its equivalence class is). The composition operation on equivalence classes is defined by first defining composition of forms and then showing that this induces a well-defined operation on classes.
"Composition" can also refer to a binary operation on representations of integers by forms. This operation is substantially more complicated{{
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>
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