# Kummer's theorem

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).

## Statement

Kummer's theorem states that for given integers n ≥ m ≥ 0 and a prime number p, the p-adic valuation ${\displaystyle \nu _{p}\left({\tbinom {n}{m}}\right)}$ is equal to the number of carries when m is added to n − m in base p.

It can be proved by writing ${\displaystyle {\tbinom {n}{m}}}$ as ${\displaystyle {\tfrac {n!}{m!(n-m)!}}}$ and using Legendre's formula.

## Multinomial coefficient generalization

Kummer's theorem may be generalized to multinomial coefficients ${\displaystyle {\tbinom {n}{m_{1},\ldots ,m_{k}}}:={\tfrac {n!}{m_{1}!\cdots m_{k}!}}}$ as follows: Write the base-${\displaystyle p}$ expansion of an integer ${\displaystyle n}$ as ${\displaystyle n=n_{0}+n_{1}p+n_{2}p^{2}+\cdots +n_{r}p^{r}}$, and define ${\displaystyle S_{p}(n)=n_{0}+n_{1}+\cdots +n_{r}}$ to be the sum of the base-${\displaystyle p}$ digits. Then

${\displaystyle \nu _{p}\left({\dbinom {n}{m_{1},\ldots ,m_{k}}}\right)={\dfrac {1}{p-1}}\left(\sum _{i=1}^{k}S_{p}(m_{i})-S_{p}(n)\right)}$