Schur's Theorem states that for any finite coloring of the positive integers, there exist three integers x, y and z of the same color satisfying x plus y equals z. Proved by Issai Schur, it is an early result in Ramsey theory that Schur originally developed while studying a version of Fermat's Last Theorem reduced modulo a prime.
Facts
StatementThe theorem states that for every positive integer c there is a positive integer S such that any partition of the integers from 1 to S into c parts puts some three integers x, y and z with x plus y equals z into the same part. 1 Classification
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Sources
1. Schur's Theorem (Ramsey Theory) (Wikipedia)
Wikimedia FoundationWikipedia, Schur's theorem, Ramsey theory sectionQuote, Wikipedia, Schur's theorem, Ramsey theory section
In Ramsey theory, Schur's theorem states that for every positive integer c, there exists a positive integer S, such that for every partition of the integers {1,…,S} into c parts, one of the parts contains integers x, y and z with x+y=z.
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