Key words and phrases: Binary quadratic forms, ideals, cycles of forms, [2] Buell, D. A., Binary Quadratic Forms, Clasical Theory and Modern Computations. “form” we mean an indefinite binary quadratic form with discriminant not a .. [1] D. A. Buell, Binary quadratic forms: Classical theory and modern computations. Citation. Lehmer, D. H. Review: D. A. Buell, Binary quadratic forms, classical theory and applications. Bull. Amer. Math. Soc. (N.S.) 23 (), no. 2,

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Binary quadratic form

A third definition is a special case of the genus of a quadratic form in n variables. His introduction of reduction allowed the quick enumeration of the classes of given discriminant and foreshadowed the eventual development of infrastructure. When f is definite, the group is finite, and when f is indefinite, it is infinite and cyclic. This article is entirely devoted to integral binary quadratic forms. Combined, the novelty and complexity made Section V notoriously difficult.

July Learn how quadratiic when to qyadratic this template message. We saw instances of this in the examples above: Another ancient problem involving quadratic forms asks us to solve Pell’s equation. There is circumstantial evidence of protohistoric knowledge of algebraic identities involving binary quadratic forms.

Binary quadratic form – Wikipedia

He replaced Lagrange’s equivalence with the more precise notion of proper equivalence, and this enabled him to show that the primitive classes of given discriminant form a group under the composition operation. Terminology has arisen for classifying classes and their forms in terms of their invariants.


Articles lacking in-text citations from July All articles lacking in-text citations All articles with unsourced statements Articles with unsourced statements from March In the first case, the sixteen representations were explicitly described. It follows that the quadratic forms are partitioned into equivalence classes, called classes of quadratic forms.

Since Gauss it has been recognized that this definition is inferior to that given above. For binary quadratic forms with other coefficients, see quadratic form. We will consider such operations in a separate section below.

Gauss introduced a very general version of a composition operator that allows composing even forms of different discriminants and imprimitive forms. Please help to improve this article by introducing more precise citations. Iterating this matrix action, we find that the infinite set of representations of 1 by f that were determined above are all equivalent.

A class invariant can mean either a function defined on equivalence classes of forms or a property shared by all forms in the same class.

There are only a finite number of pairs satisfying this constraint. We present here Arndt’s method, because it remains rather general while being simple enough to be amenable to computations by hand. This includes numerous results about quadratic number fields, which can often be translated into the language of binary quadratic forms, but also includes developments about forms themselves or that originated by thinking about forms, including Shanks’s infrastructure, Zagier’s reduction algorithm, Conway’s topographs, and Bhargava’s reinterpretation quzdratic composition through Bhargava cubes.

Lehmer : Review: D. A. Buell, Binary quadratic forms, classical theory and applications

The oldest problem in the theory of binary quadratic forms is the representation problem: From Wikipedia, the free encyclopedia. This page was last edited on 8 Novemberat Pell’s equation was already considered by the Indian mathematician Brahmagupta in the 7th century CE.


This states that forms are in the same genus if they are locally equivalent at all rational primes including the Archimedean place. The prime examples are the solution of Pell’s equation and the representation of integers as sums of two squares. Binary quadratic forms are closely related to qkadratic 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 quadrtic discriminant.

InZagier published an alternative reduction algorithm which has found several uses as an alternative to Gauss’s. For example, the matrix.

The equivalence relation above then arises from the general theory of group actions. He described an algorithm, called reductionfor constructing a canonical representative in each class, the reduced formwhose coefficients are the smallest in a suitable sense.

One way to make this a well-defined operation is to make an arbitrary convention for how to choose B —for instance, choose B to be the smallest positive solution to the system of congruences above. When the coefficients can be arbitrary complex numbersmost results are not specific to the case of two variables, so they are described in quadratic form.