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

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