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). 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 ν p ( ( n m ) ) {displaystyle u _{p}left({ binom {n}{m}} ight)} is equal to the number of carries when m is added to n − m in base p. It can be proved by writing ( n m ) {displaystyle { binom {n}{m}}} as n ! m ! ( n − m ) ! {displaystyle { frac {n!}{m!(n-m)!}}} and using Legendre's formula. Kummer's theorem may be generalized to multinomial coefficients ( n m 1 , … , m k ) := n ! m 1 ! ⋯ m k ! {displaystyle { binom {n}{m_{1},ldots ,m_{k}}}:={ frac {n!}{m_{1}!cdots m_{k}!}}} as follows: Write the base- p {displaystyle p} expansion of an integer n {displaystyle n} as n = n 0 + n 1 p + n 2 p 2 + ⋯ + n r p r {displaystyle n=n_{0}+n_{1}p+n_{2}p^{2}+cdots +n_{r}p^{r}} , and define S p ( n ) = n 0 + n 1 + ⋯ + n r {displaystyle S_{p}(n)=n_{0}+n_{1}+cdots +n_{r}} to be the sum of the base- p {displaystyle p} digits. Then