# On the divisibility of binomial coefficients

## TL;DR

This paper investigates conditions under which binomial coefficients are divisible by at least one of two primes for all intermediate values, and proves infinitely many multiples of any number have this property.

## Contribution

It provides new criteria for the divisibility of binomial coefficients by primes and extends the problem to multiple primes, including an infinite multiples result.

## Key findings

- Identifies conditions for n to have the prime divisibility property
- Extends the problem to more than two primes
- Proves infinitely many multiples of any n satisfy the property

## Abstract

In Pacific J. Math. 292 (2018), 223-238, Shareshian and Woodroofe asked if for every positive integer $n$ there exist primes $p$ and $q$ such that, for all integers $k$ with $1 \leq k \leq n-1$, the binomial coefficient $\binom{n}{k}$ 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.