On a conjecture of Graham on the p-divisibility of central binomial coefficients

Abstract

We show that for every $r \geq 1$, and all $r$ distinct (sufficiently large) primes $p_1,..., p_r > p_0(r)$, there exist infinitely many integers $n$ such that ${2n \choose n}$ is divisible by these primes to only low multiplicity. From a theorem of Kummer, an upper bound for the number of times that a prime $p_j$ can divide ${2n \choose n}$ is $1+\log n / \log p_j$; and our theorem shows that for every $\varepsilon > 0$, $r \geq 1$, and any sufficiently large primes $p_1,...,p_r > p_0(\varepsilon,r)$, we can find integers $n$ where for $j=1,...,r$, $p_j$ divides ${2n \choose n}$ with multiplicity at most $\varepsilon \log n/\log p_j$. We connect this result to a famous conjecture by R. L. Graham on whether there are infinitely many integers $n$ such that ${2n \choose n}$ is coprime to $105$.