Kummer's theorem
In mathematics, Kummer's theorem for binomial coefficients gives the p-adic valuation of a binomial coefficient, i.e., the highest power of a prime number p dividing this binomial coefficient. The theorem is named after Ernst Kummer, who proved it in the paper Kummer (1852).
Statement
Kummer's theorem states that for given integers n ≥ m ≥ 0 and a prime number p, the p-adic valuation is equal to the number of carries when m is added to n − m in base p.
It can be proved by writing as and using Legendre's formula.
See also
References
- Kummer, Ernst (1852). "Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen". Journal für die reine und angewandte Mathematik. 44: 93–146. doi:10.1515/crll.1852.44.93.
- Kummer's theorem at PlanetMath.org.
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