In
mathematics, in
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...
, Gauss composition law is a rule, invented by
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
, for performing a
binary operation on
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along wit ...
binary quadratic form
In mathematics, a binary quadratic form is a quadratic homogeneous polynomial in two variables
: q(x,y)=ax^2+bxy+cy^2, \,
where ''a'', ''b'', ''c'' are the coefficients. When the coefficients can be arbitrary complex numbers, most results are ...
s (IBQFs). Gauss presented this rule in his ''
Disquisitiones Arithmeticae
The (Latin for "Arithmetical Investigations") is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24. It is notable for having had a revolutionary impact on th ...
'', a textbook on
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...
published in 1801, in Articles 234 - 244. Gauss composition law is one of the deepest results in the theory of IBQFs and Gauss's formulation of the law and the proofs its properties as given by Gauss are generally considered highly complicated and very difficult. Several later mathematicians have simplified the formulation of the composition law and have presented it in a format suitable for numerical computations. The concept has also found generalisations in several directions.
Integral binary quadratic forms
An expression of the form
, where
are all
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s, is called an integral binary quadratic form (IBQF). The form
is called a primitive IBQF if
are relatively prime. The quantity
is called the discriminant of the IBQF
. An integer
is the discriminant of some IBQF if and only if
.
is called a
fundamental discriminant In mathematics, a fundamental discriminant ''D'' is an integer invariant (mathematics), invariant in the theory of integer, integral binary quadratic forms. If is a quadratic form with integer coefficients, then is the discriminant of ''Q''(''x'', ...
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is b ...
one of the following statements holds
*
and is
square-free {{no footnotes, date=December 2015
In mathematics, a square-free element is an element ''r'' of a unique factorization domain ''R'' that is not divisible by a non-trivial square. This means that every ''s'' such that s^2\mid r is a unit of ''R''.
A ...
,
*
where
and
is square-free.
If
and
then
is said to be positive definite; if
and
then
is said to be negative definite; if
then
is said to be indefinite.
Equivalence of IBQFs
Two IBQFs
and
are said to be equivalent (or, properly equivalent) if there exist integers α, β, γ, δ such that
:
and
The notation
is used to denote the fact that the two forms are equivalent. The relation "
" is an equivalence relation in the set of all IBQFs. The equivalence class to which the IBQF
belongs is denoted by