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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InZagier published an alternative reduction algorithm which has found several uses as an alternative to Gauss’s. We see vinary its first coefficient is well-defined, but the other two depend on the choice of B and C. Iterating this matrix action, we find that the infinite set of representations of 1 by f that were determined above are all equivalent.

Combined, the novelty and complexity made Section V notoriously difficult. Gauss also considered a coarser notion of equivalence, with each coarse class called a genus of forms. A quadratic form with integer coefficients is called an integral binary quadratic formoften abbreviated to binary buel form. There are only a finite number of pairs satisfying this constraint. If there is a need to distinguish, sometimes forms are called properly equivalent using the definition above and improperly equivalent if they are equivalent in Lagrange’s sense.

Changing signs of x and y in a solution gives another solution, so it is enough to seek just solutions in positive integers. This article includes a list of referencesbut its sources remain unclear because it has insufficient inline citations.


It follows that the quadratic forms are partitioned into equivalence classes, called classes of quadratic forms. There is a closed formula [3].

The notion of equivalence of forms can be extended to equivalent representations. Gauss introduced a very general version of a composition operator that allows composing even forms of different discriminants and imprimitive forms. Binary quadratic forms are closely related to ideals in quadratic fields, this allows the class number of a quadratic field to be calculated by qiadratic the number of reduced binary quadratic forms of a given discriminant.

Each genus is the union of a finite number of equivalence classes of the same discriminant, with the number of classes depending only on the discriminant.

Pell’s equation was already considered by the Indian mathematician Brahmagupta in the 7th century CE. 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 of composition through Bhargava cubes.

We will consider such operations in a separate section below.

Binary Quadratic Forms by Buell, Duncan a

Class groups have since become one of the central ideas in algebraic number theory. Views Read Edit View history. A variety of definitions of composition of forms has been given, often in an attempt to simplify the extremely technical and general definition of Gauss.

From Wikipedia, the free encyclopedia. We present here Arndt’s method, because it remains rather general while being simple enough to be amenable to computations by hand.

Please help to improve this article by introducing more precise citations. 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.


In matrix terminology, which is used occasionally below, when. Alternatively, we may view the result of composition, not as a form, but as an equivalence class of forms modulo the action of the group of matrices of the form. An alternative definition is described at Bhargava cubes.

Binary quadratic form

The third edition of this work includes two supplements by Dedekind. In all, there are sixteen different solution pairs. Even so, work on binary quadratic forms with integer beull continues to the present.

For binary quadratic forms with other coefficients, see quadratic form. These quadrtaic of Gauss strongly influenced both the arithmetical theory of quadratic forms in more than two variables and the subsequent development of algebraic number theory, where quadratic fields are replaced with more general number fields.

By using this site, you agree to the Terms of Use and Privacy Policy. 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. This recursive description was discussed in Theon of Smyrna’s commentary on Euclid’s Elements.