On the divisibility of binomial coefficients

Abstract

Shareshian and Woodroofe asked if for every positive integer n there exist primes p and q such that, for all integers k with 1 ≤ k ≤ n − 1, the binomial coefficient (n choose k) is divisible by at least one of p or q. We give conditions under which a number n has this property and discuss a variant of this problem involving more than two primes. We prove that every positive integer n has infinitely many multiples with this property.