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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.
Note that there is a close relation between reducing binary quadratic forms and continued fraction expansion; one step in the continued fraction expansion of a certain quadratic irrationality gives a unary operation on the set of reduced forms, which cycles through all reduced forms in one equivalence class.
A mapping q : M → R : v ↦ b(v, v) is the associated quadratic form of b, and B : M × M → R : (u, v) ↦ q(u + v) − q(u) − q(v) is the polar form of q. A quadratic form q : M → R may be characterized in the following equivalent ways: There exists an R-bilinear form b : M × M → R such that q(v) is the associated quadratic form.
The equivalence classes of quadratic irrationalities are then in bijection with the equivalence classes of binary quadratic forms, and Lagrange showed that there are finitely many equivalence classes of binary quadratic forms of given discriminant.
In mathematics, in number theory, Gauss composition law is a rule, invented by Carl Friedrich Gauss, for performing a binary operation on integral binary quadratic forms (IBQFs). Gauss presented this rule in his Disquisitiones Arithmeticae , [ 1 ] a textbook on number theory published in 1801, in Articles 234 - 244.
The discriminant of a quadratic form is invariant under linear changes of variables (that is a change of basis of the vector space on which the quadratic form is defined) in the following sense: a linear change of variables is defined by a nonsingular matrix S, changes the matrix A into , and thus multiplies the discriminant by the square of ...
Vieta jumping is a classical method in the theory of quadratic Diophantine equations and binary quadratic forms.For example, it was used in the analysis of the Markov equation back in 1879 and in the 1953 paper of Mills.
An expression of the form (,) = + +, where a, b and c are fixed integers and x and y are variable integers, is called an integer binary quadratic form. The discriminant of the form is defined as