Kummer's theorem

In mathematics, Kummer's theorem is a formula for the exponent of the highest power of a prime number p that divides a given binomial coefficient. In other words, it gives the p-adic valuation of a 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.[1]

Examples

To compute the largest power of 2 dividing the binomial coefficient ${\displaystyle {\binom {10}{3}}}$ write m = 3 and n − m = 7 in base p = 2 as 3 = 11₂ and 7 = 111₂. Carrying out the addition 11₂ + 111₂ = 1010₂ in base 2 requires three carries. And the largest power of 2 that divides ${\displaystyle {\binom {10}{3}}=120=2^{3}\cdot 15}$ is 2³.

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

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

See also

• Lucas's theorem

References

1. Mihet, Dorel (December 2010). "Legendre's and Kummer's Theorems Again". Resonance. 15 (12): 1111–1121.

• Kummer, Ernst (1852). "Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen". Journal für die reine und angewandte Mathematik. 1852 (44): 93–146. doi:10.1515/crll.1852.44.93.
• Kummer's theorem at PlanetMath.org.