We provide a way to modify and to extend a previously established inequality
by P. Erd\H{o}s, R. Graham and others and to answer a conjecture posed in the
nineties by R. Graham, which bears on the lack of divisibility of the central
binomial coefficient by three distinct, fixed odd primes. In fact the result
will show by using an approach similar to their own which they proved for the
case of two fixed odd primes, that the central binomial coefficient is not
divisible infinitely often by three distinct and fixed odd primes. Therefore a
generalization to more fixed odd primes than three but finite in number might
be possible, at least if one is able to find some sufficient condition. The
author hopes to answer this latter question in a subsequent paper.