In mathematics, Kummer's theorem for binomial coefficients gives the p-adic valuation
ν
p
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).
Kummer's theorem states that for given integers n ≥ m ≥ 0 and a prime number p, the p-adic valuation
ν
p
(
(
n
m
)
)
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
)
as
n
!
m
!
(
n
−
m
)
!
and using Legendre's formula.
