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.
