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 1852 (Kummer 1852).

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

An equivalent formation of the theorem is as follows:

Write the base-${\displaystyle p}$ expansion of the 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

${\displaystyle \nu _{p}\!{\binom {n}{m}}={\dfrac {S_{p}(m)+S_{p}(n-m)-S_{p}(n)}{p-1}}.}$

The theorem can be proved by writing ${\displaystyle {\tbinom {n}{m}}}$ as ${\displaystyle {\tfrac {n!}{m!(n-m)!}}}$ and using Legendre's formula.^[1]

To compute the largest power of 2 dividing the binomial coefficient ${\displaystyle {\tbinom {10}{3}}}$ 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:

${\displaystyle {\begin{array}{ccccc}&_{1}&_{1}&_{1}\\&&&1&1\,_{2}\\+&&1&1&1\,_{2}\\\hline &1&0&1&0\,_{2}\end{array}}}$

Therefore the largest power of 2 that divides ${\displaystyle {\tbinom {10}{3}}=120=2^{3}\cdot 15}$ is 3.

Alternatively, the form involving sums of digits can be used. The sums of digits of 3, 7, and 10 in base 2 are ${\displaystyle S_{2}(3)=1+1=2}$, ${\displaystyle S_{2}(7)=1+1+1=3}$, and ${\displaystyle S_{2}(10)=1+0+1+0=2}$ respectively. Then

${\displaystyle \nu _{2}\!{\binom {10}{3}}={\dfrac {S_{2}(3)+S_{2}(7)-S_{2}(10)}{2-1}}={\dfrac {2+3-2}{2-1}}=3.}$

Kummer's theorem can be generalized to multinomial coefficients ${\displaystyle {\tbinom {n}{m_{1},\ldots ,m_{k}}}={\tfrac {n!}{m_{1}!\cdots m_{k}!}}}$ as follows:

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

• Lucas's theorem

• 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.