When the coefficients can be arbitrary complex numbers, most results are not specific to the case of two variables, so they are described in quadratic form.
This choice is motivated by their status as the driving force behind the development of algebraic number theory.
Another ancient problem involving quadratic forms asks us to solve Pell's equation.
This recursive description was discussed in Theon of Smyrna's commentary on Euclid's Elements.
The number of representations of an integer n by a form f is finite if f is definite and infinite if f is indefinite.
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
A complete set of representatives for these classes can be given in terms of reduced forms defined in the section below.
This operation is substantially more complicated[citation needed] than composition of forms, but arose first historically.
We present here Arndt's method, because it remains rather general while being simple enough to be amenable to computations by hand.
We see that its first coefficient is well-defined, but the other two depend on the choice of B and C. 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.
Thus, composition gives a well-defined function from pairs of binary quadratic forms to such classes.
, and as mentioned above, Gauss showed these classes form a finite abelian group.
Gauss also considered a coarser notion of equivalence, with each coarse class called a genus of forms.
This states that forms are in the same genus if they are locally equivalent at all rational primes (including the Archimedean place).
There is circumstantial evidence of protohistoric knowledge of algebraic identities involving binary quadratic forms.
The prime examples are the solution of Pell's equation and the representation of integers as sums of two squares.
Several centuries later, his ideas were extended to a complete solution of Pell's equation known as the chakravala method, attributed to either of the Indian mathematicians Jayadeva or Bhāskara II.
[8] The problem of representing integers by sums of two squares was considered in the 3rd century by Diophantus.
Lagrange was the first to realize that "a coherent general theory required the simulatenous consideration of all forms.
"[12] He was the first to recognize the importance of the discriminant and to define the essential notions of equivalence and reduction, which, according to Weil, have "dominated the whole subject of quadratic forms ever since".
His introduction of reduction allowed the quick enumeration of the classes of given discriminant and foreshadowed the eventual development of infrastructure.
In 1798, Legendre published Essai sur la théorie des nombres, which summarized the work of Euler and Lagrange and added some of his own contributions, including the first glimpse of a composition operation on forms.
The theory was vastly extended and refined by Gauss in Section V of Disquisitiones Arithmeticae.
He introduced genus theory, which gives a powerful way to understand the quotient of the class group by the subgroup of squares.
(Gauss and many subsequent authors wrote 2b in place of b; the modern convention allowing the coefficient of xy to be odd is due to Eisenstein).
These investigations 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.
Section V of Disquisitiones contains truly revolutionary ideas and involves very complicated computations, sometimes left to the reader.
Dirichlet published simplifications of the theory that made it accessible to a broader audience.
Even so, work on binary quadratic forms with integer coefficients continues to the present.