On the integrality of the elementary symmetric functions of $1, 1/3, ..., 1/(2n-1)$

TL;DR

This paper proves that for all integers n ≥ 2, none of the elementary symmetric functions of the sequence 1, 1/3, ..., 1/(2n-1) are integers, extending previous results on related sequences.

Contribution

It establishes a new non-integrality result for elementary symmetric functions of a specific sequence of reciprocals, generalizing prior findings.

Findings

Elementary symmetric functions of 1, 1/3, ..., 1/(2n-1) are not integers for n ≥ 2

Extends previous results on the non-integrality of symmetric functions of reciprocals

Provides a new proof technique for non-integrality in specific rational sequences

Abstract

Erdos and Niven proved that for any positive integers $m$ and $d$, there are only finitely many positive integers $n$ for which one or more of the elementary symmetric functions of $1/m,1/(m+d), ..., 1/(m+nd)$ are integers. Recently, Chen and Tang proved that if $n\ge 4$, then none of the elementary symmetric functions of $1,1/2, ..., 1/n$ is an integer. In this paper, we show that if $n\ge 2$, then none of the elementary symmetric functions of $1, 1/3, ..., 1/(2n-1)$ is an integer.