Prime power divisors of binomial coefficients

Abstract

It is known that for sufficiently large n and m and any r the binomial coefficient ( n m ) which is close to the middle coefficient is divisible by p r where p is a ‘large’ prime. We prove the exact divisibility of ( n m ) by p r for p > c( n). The lower bound is essentially the best possible. We also prove some other results on divisibility of binomial coefficients.