Kummer's theorem

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In mathematics, Kummer's theorem for binomial coefficients gives the p-adic valuation of a binomial coefficient, i.e., the exponent of 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).


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.

Multinomial coefficient generalization[edit]

Kummer's theorem may be generalized to multinomial coefficients as follows: Write the base- expansion of an integer as , and define to be the sum of the base- digits. Then


See also[edit]


  • 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.