Schur's theorem

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. 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. 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. Using this definition, the first few Schur numbers are S(1) = 2, 5, 14, 45, 161, ... (OEIS: A030126) The proof that S(5) = 161 was announced in 2017 and took up 2 petabytes of space. 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 { a 1 , … , a n } {displaystyle {a_{1},ldots ,a_{n}}} is a set of integers such that gcd ( a 1 , … , a n ) = 1 {displaystyle gcd(a_{1},ldots ,a_{n})=1} , the number of different tuples of non-negative integer numbers ( c 1 , … , c n ) {displaystyle (c_{1},ldots ,c_{n})} such that x = c 1 a 1 + ⋯ + c n a n {displaystyle x=c_{1}a_{1}+cdots +c_{n}a_{n}} when x {displaystyle x} goes to infinity is: As a result, for every set of relatively prime numbers { a 1 , … , a n } {displaystyle {a_{1},ldots ,a_{n}}} there exists a value of x {displaystyle x} such that every larger number is representable as a linear combination of { a 1 , … , a n } {displaystyle {a_{1},ldots ,a_{n}}} 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 differential geometry, Schur's theorem compares the distance between the endpoints of a space curve C ∗ {displaystyle C^{*}} to the distance between the endpoints of a corresponding plane curve C {displaystyle C} of less curvature.