Lack of Divisibility of ${2N \choose N}$ by three fixed odd primes infinitely often, through the Extension of a Result by P. Erd\H{o}s, et al

TL;DR

This paper extends previous results to demonstrate that the central binomial coefficient ${2N race N}$ is not divisible infinitely often by three fixed odd primes, generalizing earlier work from two primes to three and potentially more.

Contribution

It introduces a modified inequality extending Erd"H{o}s and Graham's work, proving non-divisibility of ${2N race N}$ by three fixed odd primes infinitely often.

Findings

Central binomial coefficient not divisible infinitely often by three fixed odd primes

Extension of inequality from two to three primes

Potential for further generalization to more primes

Abstract

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.