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

Ernie Croot; Hamed Mousavi; Maxie Schmidt

paper

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

arXiv:2201.11274

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 $(n 2 n)$ 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 $(n 2 n)$ is $1 + lo g n / lo g p_{j}$ ; and our theorem shows that for every $ε > 0$ , $r \geq 1$ , and any sufficiently large primes $p_{1}, ..., p_{r} > p_{0} (ε, r)$ , we can find integers $n$ where for $j = 1, ..., r$ , $p_{j}$ divides $(n 2 n)$ with multiplicity at most $ε lo g n / lo g p_{j}$ . We connect this result to a famous conjecture by R. L. Graham on whether there are infinitely many integers $n$ such that $(n 2 n)$ is coprime to $105$ .

The main theorem was corrected from previous draft. Several more corrections and simplifications were made