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Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a quadratic form is a homogeneous polynomial

Homogeneous polynomial

In mathematics, a homogeneous polynomial is a polynomial whose monomials with nonzero coefficients all have thesame total degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial...

of degree

Degree of a polynomial

The degree of a polynomial represents the highest degree of a polynominal's terms , should the polynomial be expressed in canonical form . The degree of an individual term is the sum of the exponents acting on the term's variables...

two in a number of variables. For example,

is a quadratic form in the variables x and y.

Quadratic forms occupy a central place in various branches of mathematics, including number theory

Number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

, linear algebra

Linear algebra

Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...

, group theory

Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...

(orthogonal group

Orthogonal group

In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...

), differential geometry (Riemannian metric), differential topology

Differential topology

In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...

(intersection forms

Intersection form (4-manifold)

In mathematics, the intersection form of an oriented compact 4-manifold is a special symmetric bilinear form on the 2nd cohomology group of the 4-manifold...

of four-manifolds), and Lie theory

Lie theory

Lie theory is an area of mathematics, developed initially by Sophus Lie.Early expressions of Lie theory are found in books composed by Lie with Friedrich Engel and Georg Scheffers from 1888 to 1896....

(the Killing form

Killing form

In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras...

Introduction

Quadratic forms are homogeneous quadratic polynomials in n variables.
In the cases of one, two, and three variables they are called unary, binary

Binary quadratic form

In mathematics, a binary quadratic form is a quadratic form in two variables. More concretely, it is a homogeneous polynomial of degree 2 in two variableswhere a, b, c are the coefficients...

, and ternary and have the following explicit form:

,

where a,…,f are the coefficients. Note that quadratic function

Quadratic function

A quadratic function, in mathematics, is a polynomial function of the formf=ax^2+bx+c,\quad a \ne 0.The graph of a quadratic function is a parabola whose axis of symmetry is parallel to the y-axis....

s, such as ax²+bx+c in the one variable case, are not quadratic forms, as they are typically not homogeneous

Homogeneous polynomial

In mathematics, a homogeneous polynomial is a polynomial whose monomials with nonzero coefficients all have thesame total degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial...

(unless b and c are both 0).

The theory of quadratic forms and methods used in their study depend in a large measure on the nature of the coefficients, which may be real

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

or complex

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

numbers, rational number

Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

s, or integer

Integer

The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

s. In linear algebra

Linear algebra

, analytic geometry

Analytic geometry

Analytic geometry, or analytical geometry has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties...

, and in the majority of applications of quadratic forms, the coefficients are real or complex numbers. In the algebraic theory of quadratic forms, the coefficients are elements of a certain field. In the arithmetic theory of quadratic forms, the coefficients belong to a fixed commutative ring

Commutative ring

In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....

, frequently the integers Z or the p-adic integers Z_p. Binary quadratic forms have been extensively studied in number theory

Number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

, in particular, in the theory of quadratic field

Quadratic field

In algebraic number theory, a quadratic field is an algebraic number field K of degree two over Q. It is easy to show that the map d ↦ Q is a bijection from the set of all square-free integers d ≠ 0, 1 to the set of all quadratic fields...

s, continued fraction

Continued fraction

In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...

s, and modular forms. The theory of integral quadratic forms in n variables has important applications to algebraic topology

Algebraic topology

Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...

.

Using homogeneous coordinates

Homogeneous coordinates

In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry much as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points,...

, a non-zero quadratic form in n variables defines an (n−2)-dimensional quadric

Quadric (projective geometry)

In projective geometry a quadric is the set of points of a projective space where a certain quadratic form on the homogeneous coordinates becomes zero...

in the (n−1)-dimensional projective space

Projective space

In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....

. This is a basic construction in projective geometry

Projective geometry

In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...

. In this way one may visualize 3-dimensional real quadratic forms as conic sections.

A closely related notion with geometric overtones is a quadratic space, which is a pair (V,q), with V a vector space

Vector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

over a field k, and q:V → k a quadratic form on V. An example is given by the three-dimensional Euclidean space

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

and the square of the Euclidean norm expressing the distance

Distance

Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria . In mathematics, a distance function or metric is a generalization of the concept of physical distance...

between a point with coordinates (x,y,z) and the origin:

History

The study of particular quadratic forms, in particular the question of whether a given integer can be the value of a quadratic form over the integers, dates back many centuries. One such case is Fermat's theorem on sums of two squares, which determines when an integer may be expressed in the form

, where

are integers. This problem is related to the problem of finding Pythagorean triple

Pythagorean triple

A Pythagorean triple consists of three positive integers a, b, and c, such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer k. A primitive Pythagorean triple is one in which a, b and c are pairwise coprime...

s, which appeared in the second millennium B.C.

In 628, the Indian mathematician Brahmagupta

Brahmagupta

Brahmagupta was an Indian mathematician and astronomer who wrote many important works on mathematics and astronomy. His best known work is the Brāhmasphuṭasiddhānta , written in 628 in Bhinmal...

wrote Brahmasphutasiddhanta
Brahmasphutasiddhanta
The main work of Brahmagupta, Brāhmasphuṭasiddhānta , written c.628, contains ideas including a good understanding of the mathematical role of zero, rules for manipulating both negative and positive numbers, a method for computing square roots, methods of solving linear and some quadratic...

which includes, among many other things, a study of equations of the form

. In particular he considered what is now called Pell's equation

Pell's equation

Pell's equation is any Diophantine equation of the formx^2-ny^2=1\,where n is a nonsquare integer. The word Diophantine means that integer values of x and y are sought. Trivially, x = 1 and y = 0 always solve this equation...

, and found a method for its solution. In Europe this problem was studied by Brouncker

William Brouncker, 2nd Viscount Brouncker

William Brouncker, 2nd Viscount Brouncker, PRS was an English mathematician.Brouncker obtained a DM at the University of Oxford in 1647. He was one of the founders and the first President of the Royal Society. In 1662, he became Chancellor to Queen Catherine, then chief of the Saint Catherine's...

, Euler

Leonhard Euler

Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

and Lagrange

Joseph Louis Lagrange

Joseph-Louis Lagrange , born Giuseppe Lodovico Lagrangia, was a mathematician and astronomer, who was born in Turin, Piedmont, lived part of his life in Prussia and part in France, making significant contributions to all fields of analysis, to number theory, and to classical and celestial mechanics...

.

In 1801 Gauss

Carl Friedrich Gauss

Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...

published Disquisitiones Arithmeticae
Disquisitiones Arithmeticae
The Disquisitiones Arithmeticae 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...

, a major portion of which was devoted to a complete theory of binary quadratic form

Binary quadratic form

In mathematics, a binary quadratic form is a quadratic form in two variables. More concretely, it is a homogeneous polynomial of degree 2 in two variableswhere a, b, c are the coefficients...

s over the integer

Integer

The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

s. Since then, the concept has been generalized, and the connections with quadratic number fields, the modular group

Modular group

In mathematics, the modular group Γ is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics...

, and other areas of mathematics have been further elucidated.

Real quadratic forms

Any n×n real symmetric matrix A determines a quadratic form q_A in n variables by the formula

Conversely, given a quadratic form in n variables, its coefficients can be arranged into an n×n symmetric matrix. One of the most important questions in the theory of quadratic forms is how much can one simplify a quadratic form q by a homogeneous linear change of variables. A fundamental theorem due to Jacobi asserts that q can be brought to a diagonal form

so that the corresponding symmetric matrix is diagonal

Diagonal matrix

In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero. The diagonal entries themselves may or may not be zero...

, and this is even possible to accomplish with a change of variables given by an orthogonal matrix

Orthogonal matrix

In linear algebra, an orthogonal matrix , is a square matrix with real entries whose columns and rows are orthogonal unit vectors ....

– in this case the coefficients λ₁, λ₂, …, λ_n are in fact determined uniquely up to a permutation. If the change of variables is given by an invertible matrix, not necessarily orthogonal, then the coefficients λ_i can be made to be 0,1, and −1. Sylvester's law of inertia

Sylvester's law of inertia

Sylvester's law of inertia is a theorem in matrix algebra about certain properties of the coefficient matrix of a real quadratic form that remain invariant under a change of coordinates...

states that the numbers of 1 and −1 are invariants

Invariant (mathematics)

In mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used...

of the quadratic form, in the sense that any other diagonalization will contain them in the same quantities. The signature of the quadratic form is the triple (n₀, n₊, n₋) where n₀ is the number 0s and n_± is the number of ±1s. Sylvester's law of inertia shows that this is a well-defined quantity attached to the quadratic form. The case when all λ_i have the same sign is especially important: in this case the quadratic form is called positive definite (all 1) or negative definite (all −1); if none of the terms are 0 then the form is called ; this includes positive definite, negative definite, and indefinite (a mix of 1 and −1); equivalently, a nondegenerate quadratic form is one whose associated symmetric form is a nondegenerate bilinear form. A real vector space with an indefinite nondegenerate quadratic form of index

(p 1s, q −1s) is often denoted as

particularly in the physical theory of space-time.

Below we reformulate these results in a different way.

Let q be a quadratic form defined on an n-dimensional real

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

vector space. Let A be the matrix of the quadratic form q in a given basis. This means that A is a symmetric n×n matrix such that

where x is the column vector of coordinates of v in the chosen basis. Under a change of basis, the column x is multiplied on the left by an n×n invertible matrix S, and the symmetric square matrix A is transformed into another symmetric square matrix B of the same size according to the formula

Any symmetric matrix A can be transformed into a diagonal matrix

by a suitable choice of an orthogonal matrix S, and the diagonal entries of B are uniquely determined — this is Jacobi's theorem. If S is allowed to be any invertible matrix then B can be made to have only 0,1, and −1 on the diagonal, and the number of the entries of each type (n₀ for 0, n₊ for 1, and n₋ for −1) depends only on A. This is one of the formulations of Sylvester's law of inertia and the numbers n₊ and n₋ are called the positive and negative indices of inertia. Although their definition involved a choice of basis and consideration of the corresponding real symmetric matrix A, Sylvester's law of inertia means that they are invariants of the quadratic form q.

The quadratic form q is positive definite (resp., negative definite) if q(v)>0 (resp., q(v)<0) for every nonzero vector v. When q(v) assumes both positive and negative values, q is an indefinite quadratic form. The theorems of Jacobi and Sylvester show that any positive definite quadratic form in n variables can be brought to the sum of n squares by a suitable invertible linear transformation: geometrically, there is only one positive definite real quadratic form of every dimension. Its isometry group

Isometry group

In mathematics, the isometry group of a metric space is the set of all isometries from the metric space onto itself, with the function composition as group operation...

is a compact

Compact space

In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

orthogonal group

Orthogonal group

In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...

O(n). This stands in contrast with the case of indefinite forms, when the corresponding group, the indefinite orthogonal group O(p,q), is non-compact. Further, the isometry groups of Q and −Q are the same (

), but the associated Clifford algebra

Clifford algebra

In mathematics, Clifford algebras are a type of associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal...

s (and hence Pin group

Pin group

In mathematics, the pin group is a certain subgroup of the Clifford algebra associated to a quadratic space. It maps 2-to-1 to the orthogonal group, just as the spin group maps 2-to-1 to the special orthogonal group....

s) are different.

Definitions

An n-ary quadratic form over a field K is a homogeneous polynomial

Homogeneous polynomial

In mathematics, a homogeneous polynomial is a polynomial whose monomials with nonzero coefficients all have thesame total degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial...

of degree 2 in n variables with coefficients in K:

This formula may be rewritten using matrices: let x be the column vector with components x₁, …, x_n and A = (a_ij) be the n×n matrix over K whose entries are the coefficients of q. Then

Two n-ary quadratic forms φ and ψ over K are equivalent if there exists a nonsingular linear transformation T ∈GL_n(K) such that

Let us assume that the characteristic of K is different from 2.
(The theory of quadratic forms over a field of characteristic 2 has important differences and many definitions and theorems have to be modified.) The coefficient matrix A of q may be replaced by the symmetric matrix 1/2(A + A^T) with the same quadratic form, so it may be assumed from the outset that A is symmetric. Moreover, a symmetric matrix A is uniquely determined by the corresponding quadratic form. Under an equivalence T, the symmetric matrix A of φ and the symmetric matrix B of ψ are related as follows:

The associated bilinear form of a quadratic form q is defined by

Thus, b_q is a symmetric bilinear form

Symmetric bilinear form

A symmetric bilinear form is a bilinear form on a vector space that is symmetric. Symmetric bilinear forms are of great importance in the study of orthogonal polarity and quadrics....

over K with matrix A. Conversely, any symmetric bilinear form b defines a quadratic form

and these two processes are the inverses of one another. As a consequence, over a field of characteristic not equal to 2, the theories of symmetric bilinear forms and of quadratic forms in n variables are essentially the same.

Quadratic spaces

A quadratic form q in n variables over K induces a map from the n-dimensional coordinate space Kⁿ into K:

The map Q is a quadratic map, which means that it has the properties:

The map B_Q: V×V → K defined below is bilinear over K:

The pair (V,Q) consisting of a finite-dimensional vector space V over K and a quadratic map from V to K is called a quadratic space and B_Q is the associated bilinear form of Q. The notion of a quadratic space is a coordinate-free version of the notion of quadratic form. Sometimes, Q is also called a quadratic form.

Two n-dimensional quadratic spaces (V,Q) and (V ′, Q ′) are isometric if there exists an invertible linear transformation T: V →V ′ (isometry) such that

The isometry classes of n-dimensional quadratic spaces over K correspond to the equivalence classes of n-ary quadratic forms over K.

Further definitions

Two elements v and w of V are called orthogonal if B(v, w)=0. The kernel of a bilinear form B consists of the elements that are orthogonal to each elements of V. Q is non-singular if the kernel of its associated bilinear form is 0. If there exists a non-zero v in V such that Q(v) = 0, the quadratic form Q is isotropic

Isotropic quadratic form

In mathematics, a quadratic form over a field F is said to be isotropic if there is a non-zero vector on which it evaluates to zero. Otherwise the quadratic form is anisotropic. More precisely, if q is a quadratic form on a vector space V over F, then a non-zero vector v in V is said to be...

, otherwise it is anisotropic. This terminology also applies to vectors and subspaces of a quadratic space. If the restriction of Q to a subspace U of V is identically zero, U is totally singular.

The orthogonal group

Orthogonal group

In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...

of a non-singular quadratic form Q is the group of the linear automorphisms of V that preserve Q, i.e. the group of isometries of (V, Q) into itself.

Equivalence of forms

Every quadratic form q in n variables over a field of characteristic not equal to 2 is equivalent to a diagonal form

Such a diagonal form is often denoted by

.

Classification of all quadratic forms up to equivalence can thus be reduced to the case of diagonal forms.

Geometric Meaning

If we let the equation be

with symmetric matrix A, then the geometric meaning is as follows.

If all eigenvalues of A are non-zero, then it is an ellipsoid or a hyperboloid. If all the eigenvalues are positive, then it is an ellipsoid; if all the eigenvalues are negative, it is an image ellipsoid; if some eigenvalues are positive and some are negative, then it is a hyperboloid.

If there exist one or more eigenvalues

, then if the corresponding

, it is a paraboloid

Paraboloid

In mathematics, a paraboloid is a quadric surface of special kind. There are two kinds of paraboloids: elliptic and hyperbolic. The elliptic paraboloid is shaped like an oval cup and can have a maximum or minimum point....

(either elliptic or hyperbolic); if the corresponding

, the dimension i degenerates and does not get into play, and the geometric meaning will be determined by other eigenvalues and other components of b. When it is a paraboloid, whether it is elliptic or hyperbolic is determined by whether all other non-zero eigenvalues are of the same sign: if they are, then it is elliptic; otherwise, it is hyperbolic.

Integral quadratic forms

Quadratic forms over the ring of integers are called integral quadratic forms, whereas the corresponding modules are quadratic lattices (sometimes, simply lattice

Lattice (group)

In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...

s). They play an important role in number theory

Number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

and topology

Topology

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

.

An integral quadratic form has integer coefficients, such as

; equivalently, given a lattice

in a vector space

(over a field with characteristic 0, such as

), a quadratic form

is integral with respect to

if and only if it is integer-valued on

, meaning

.

This is the current use of the term; in the past it was sometimes used differently, as detailed below.

Historical use

Historically there was some confusion and controversy over whether the notion of integral quadratic form should mean:
twos in: the quadratic form associated to a symmetric matrix with integer coefficients
twos out: a polynomial with integer coefficients (so the associated symmetric matrix may have half-integer coefficients off the diagonal)
This debate was due to the confusion of quadratic forms (represented by polynomials) and symmetric bilinear forms (represented by matrices), and "twos out" is now the accepted convention; "twos in" is instead the theory of integral symmetric bilinear forms (integral symmetric matrices).

In "twos in", binary quadratic forms are of the form

, represented by the symmetric matrix

; this is the convention Gauss

Gauss

Gauss may refer to:*Carl Friedrich Gauss, German mathematician and physicist*Gauss , a unit of magnetic flux density or magnetic induction*GAUSS , a software package*Gauss , a crater on the moon...

uses in Disquisitiones Arithmeticae

Disquisitiones Arithmeticae

The Disquisitiones Arithmeticae 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...

.

In "twos out", binary quadratic forms are of the form

, represented by the symmetric matrix

.

Several points of view mean that twos out has been adopted as the standard convention. Those include:

better understanding of the 2-adic theory of quadratic forms, the 'local' source of the difficulty;
the lattice
Lattice (group)
In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...

point of view, which was generally adopted by the experts in the arithmetic of quadratic forms during the 1950s;
the actual needs for integral quadratic form theory in topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

for intersection theory
Intersection theory
In mathematics, intersection theory is a branch of algebraic geometry, where subvarieties are intersected on an algebraic variety, and of algebraic topology, where intersections are computed within the cohomology ring. The theory for varieties is older, with roots in Bézout's theorem on curves and...

;
the Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...

and algebraic group
Algebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...

aspects.

Universal quadratic forms

A quadratic form representing all of the positive integers is sometimes called universal. Lagrange's four-square theorem

Lagrange's four-square theorem

Lagrange's four-square theorem, also known as Bachet's conjecture, states that any natural number can be represented as the sum of four integer squaresp = a_0^2 + a_1^2 + a_2^2 + a_3^2\ where the four numbers are integers...

shows that

is universal. Ramanujan generalized this to

and found 54 {a,b,c,d} such that it can generate all positive integers, namely,

{1,1,1,d}; d = 1-7

{1,1,2,d}; d = 2-14

{1,1,3,d}; d = 3-6

{1,2,2,d}; d = 2-7

{1,2,3,d}; d = 3-10

{1,2,4,d}; d = 4-14

{1,2,5,d}; d = 6-10

There are also forms that can express nearly all positive integers except one, such as {1,2,5,5} which has 15 as the exception. Recently, the 15 and 290 theorems have completely characterized universal integral quadratic forms: if all coefficients are integers, then it represents all positive integers if and only if it represents all integers up through 290; if it has an integral matrix, it represents all positive integers if and only if it represents all integers up through 15.

Introduction

History

Real quadratic forms

Definitions

Quadratic spaces

Further definitions

Equivalence of forms

Geometric Meaning

Integral quadratic forms

Historical use

Universal quadratic forms

See also