On the elementary symmetric functions of $1, 1/2, \ldots , 1/n$

TL;DR

This paper proves that for all integers n ≥ 4, none of the elementary symmetric functions of the sequence 1, 1/2, ..., 1/n are integers, resolving a problem posed by Erdős and Niven in 1946.

Contribution

It establishes that only for n < 4 can elementary symmetric functions of these reciprocals be integers, solving a long-standing open problem.

Findings

Elementary symmetric functions are non-integer for all n ≥ 4

Finiteness of n with integer symmetric functions is confirmed for n<4

Addresses a problem posed by Erdős and Niven in 1946

Abstract

In 1946, P. Erd\H os and I. Niven proved that there are only finitely many positive integers $n$ for which one or more elementary symmetric functions of $1, 1/2, \ldots , 1/n$ are integers. In this paper we solve this old problem by showing that if $n\ge 4$, then none of elementary symmetric functions of $1, 1/2, \ldots , 1/n$ is an integer.