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

Abstract

Let $$ be pairwise distinct primes. From a theorem of Kummer, each prime $$ can divide $$ at most $$ times. We show that, for all $$, if $$ are sufficiently large in terms of $$ and $$, then there exist infinitely many positive integers $$ such that each $$ divides $$ at most $$ times. We connect this result to a famous conjecture by Graham on whether there are infinitely many integers $$ such that $$ is coprime to 105.