On a nonintegrality conjecture

TL;DR

This paper investigates the conjecture that a certain sum involving binomial coefficients and rational terms is never an integer, showing that the set of n for which it could be integral is very sparse, using advanced prime distribution results.

Contribution

It proves that the set of n with integral sums for some r has asymptotic density zero, advancing understanding of the conjecture in the n aspect.

Findings

The set of n with integral sums has asymptotic density zero.

The problem is studied in the 'n aspect' rather than fixed r.

Deep results on prime distribution are employed.

Abstract

It is conjectured that the sum $$ S_r(n)=\sum_{k=1}^{n} \frac{k}{k+r}\binom{n}{k} $$ for positive integers $r,n$ is never integral. This has been shown for $r\le 22$. In this note we study the problem in the ``$n$ aspect" showing that the set of $n$ such that $S_r(n)\in {\mathbb Z}$ for some $r\ge 1$ has asymptotic density $0$. Our principal tools are some deep results on the distribution of primes in short intervals.