Kummer's Theorem gives the exact power of a prime dividing a binomial coefficient, showing it equals the number of carries produced when the two lower arguments of the binomial coefficient are added together in base that prime. Named for Ernst Kummer, it is a classical result of number theory closely related to, and often paired with, Lucas' Theorem on binomial coefficients modulo a prime.
Facts
StatementThe exact power of a prime p dividing a binomial coefficient equals the number of carries produced when the two lower numbers of the binomial coefficient are added together in base p. 1 Classification
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Source Wolfram MathWorld: Kummer's Theorem
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Source Kummer's theorem, Wikipedia
Sources
1. Kummer's theorem, Wikipedia
Lead section
The theorem is named after Ernst Kummer, who proved it in 1852 (Kummer 1852).
Lead section, naming sentence
The theorem is named after Ernst Kummer, who proved it in 1852 (Kummer 1852).
Proved By: Ernst Kummer, Lead
The theorem is named after Ernst Kummer, who proved it in 1852 (Kummer 1852).
View the SourceWolfram MathWorld: Kummer's Theorem
Wolfram MathWorldIn Branch: Analysis, Subject classificationsQuote, In Branch: Analysis, Subject classifications
Calculus and Analysis Special Functions
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