In
discrete mathematics
Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuous f ...
, Schur's theorem is any of several
theorem
In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...
s of the
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
Issai Schur
Issai Schur (10 January 1875 – 10 January 1941) was a Russian mathematician who worked in Germany for most of his life. He studied at the Humboldt University of Berlin, University of Berlin. He obtained his doctorate in 1901, became lecturer i ...
. In
differential geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
, Schur's theorem is a theorem of
Axel Schur. In
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
, Schur's theorem is often called
Schur's property, also due to Issai Schur.
Ramsey theory
In
Ramsey theory
Ramsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of the mathematical field of combinatorics that focuses on the appearance of order in a substructure given a structure of a known size. Problems in R ...
, Schur's theorem states that for any
partition of the
positive integer
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...
s into a finite number of parts, one of the parts contains three integers ''x'', ''y'', ''z'' with
:
For every positive integer ''c'', ''S''(''c'') denotes the smallest number ''S'' such that for every partition of the integers
into ''c'' parts, one of the parts contains integers ''x'', ''y'', and ''z'' with
. Schur's theorem ensures that ''S''(''c'') is well-defined for every positive integer ''c''. The numbers of the form ''S''(''c'') are called Schur's numbers.
Folkman's theorem generalizes Schur's theorem by stating that there exist arbitrarily large sets of integers, all of whose
nonempty
In mathematics, the empty set or void set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, whi ...
sums belong to the same part.
Using this definition, the only known Schur numbers are ''S''(n) 2, 5, 14, 45, and 161 () The
proof
Proof most often refers to:
* Proof (truth), argument or sufficient evidence for the truth of a proposition
* Alcohol proof, a measure of an alcoholic drink's strength
Proof may also refer to:
Mathematics and formal logic
* Formal proof, a co ...
that was announced in 2017 and required 2
petabyte
The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable un ...
s of space.
Combinatorics
In
combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...
, Schur's theorem tells the number of ways for expressing a given number as a (non-negative, integer) linear combination of a fixed set of
relatively prime
In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equiv ...
numbers. In particular, if
is a set of integers such that
, the number of different multiples of non-negative integer numbers
such that
when
goes to infinity is:
:
As a result, for every set of relatively prime numbers
there exists a value of
such that every larger number is representable as a linear combination of
in at least one way. This consequence of the theorem can be recast in a familiar context considering the problem of changing an amount using a set of coins. If the denominations of the coins are relatively prime numbers (such as 2 and 5) then any sufficiently large amount can be changed using only these coins. (See
Coin problem
In mathematics, the coin problem (also referred to as the Frobenius coin problem or Frobenius problem, after the mathematician Ferdinand Georg Frobenius, Ferdinand Frobenius) is a mathematical problem that asks for the largest monetary amount ...
.)
Differential geometry
In
differential geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
, Schur's theorem compares the distance between the endpoints of a space curve
to the distance between the endpoints of a corresponding plane curve
of less curvature.
Suppose
is a plane curve with curvature
which makes a convex curve when closed by the chord connecting its endpoints, and
is a curve of the same length with curvature
. Let
denote the distance between the endpoints of
and
denote the distance between the endpoints of
. If
then
.
Schur's theorem is usually stated for
curves, but
John M. Sullivan has observed that Schur's theorem applies to curves of finite total curvature (the statement is slightly different).
Linear algebra
In
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrix (mathemat ...
, Schur’s theorem is referred to as either the
triangularization of a
square matrix
In mathematics, a square matrix is a Matrix (mathematics), matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied.
Squ ...
with
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
entries, or of a square matrix with
real
Real may refer to:
Currencies
* Argentine real
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Nature and science
* Reality, the state of things as they exist, rathe ...
entries and real
eigenvalue
In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
s.
Functional analysis
In
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
and the study of
Banach space
In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
s, Schur's theorem, due to
I. Schur, often refers to
Schur's property, that for certain spaces,
weak convergence implies convergence in the norm.
Number theory
In
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, Issai Schur showed in 1912 that for every nonconstant
polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
''p''(''x'') with integer
coefficient
In mathematics, a coefficient is a Factor (arithmetic), multiplicative factor involved in some Summand, term of a polynomial, a series (mathematics), series, or any other type of expression (mathematics), expression. It may be a Dimensionless qu ...
s, if ''S'' is the set of all nonzero values
, then the set of
primes
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
that divide some member of ''S'' is infinite.
See also
*
Schur's lemma (from Riemannian geometry)
References
* Herbert S. Wilf (1994)
generatingfunctionology Academic Press.
*
Shiing-Shen Chern
Shiing-Shen Chern (; , ; October 26, 1911 – December 3, 2004) was a Chinese American mathematician and poet. He made fundamental contributions to differential geometry and topology. He has been called the "father of modern differential geome ...
(1967). Curves and Surfaces in Euclidean Space. In ''Studies in Global Geometry and Analysis.'' Prentice-Hall.
* Issai Schur (1912). Über die Existenz unendlich vieler Primzahlen in einigen speziellen arithmetischen Progressionen, Sitzungsberichte der Berliner Math.
Further reading
* Dany Breslauer and Devdatt P. Dubhashi (1995)
Combinatorics for Computer Scientists*
John M. Sullivan (2006)
Curves of Finite Total Curvature arXiv.
{{Functional analysis
Theorems in discrete mathematics
Ramsey theory
Additive combinatorics
Theorems in combinatorics
Theorems in differential geometry
Theorems in linear algebra
Theorems in functional analysis
Computer-assisted proofs