Schur's theorem

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search

In discrete mathematics, Schur's theorem is any of several theorems of the mathematician Issai Schur. In differential geometry, Schur's theorem is a theorem of Axel Schur. In functional analysis, Schur's theorem is often called Schur's property, also due to Issai Schur.

Ramsey theory[edit]

In Ramsey theory, Schur's theorem states that for any partition of the positive integers into a finite number of parts, one of the parts contains three integers x, y, z with

Moreover, for every positive integer c, there exists a number S(c), called Schur's number, such that for every partition of the integers

into c parts, one of the parts contains integers x, y, and z with

Folkman's theorem generalizes Schur's theorem by stating that there exist arbitrarily large sets of integers all of whose nonempty sums belong to the same part.


In combinatorics, 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 numbers. In particular, if is a set of integers such that , the number of different tuples 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.)

Differential geometry[edit]

In differential geometry, 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[edit]

In linear algebra Schur’s theorem is referred to as either the triangularization of a square matrix with complex entries, or of a square matrix with real entries and real eigenvalues.

Functional analysis[edit]

In functional analysis and the study of Banach spaces, Schur's theorem, due to J. Schur, often refers to Schur's property, that for certain spaces, weak convergence implies convergence in the norm.

Number theory[edit]

In number theory, Issai Schur showed in 1912 that for every nonconstant polynomial p(x) with integer coefficients, if S is the set of all nonzero values , then the set of primes that divide some member of S is infinite.

See also[edit]


  • Herbert S. Wilf (1994). generatingfunctionology. Academic Press.
  • Shiing-Shen Chern (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[edit]