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

Abstract

AbstractLet 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.