Binary quadratic form: Difference between revisions

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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[[representative (mathematics)|representative]]s 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|loc=}}</ref> The set of all such representations constitutes a complete set of representatives for equivalence classes of representations.

== Reduction and class numbers<!--'Class number (binary quadratic forms)' redirects here--> ==