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
Statement Form Connections
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Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Sources
1. Kummer's theorem, Wikipedia
Statement section
the p-adic valuation of the binomial coefficient is equal to the number of carries when m is added to n - m in base p.
Lead section, naming sentence
The theorem is named after Ernst Kummer, who proved it in 1852 (Kummer 1852).
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