The elementary symmetric functions of reciprocals of the elements of arithmetic progressions

TL;DR

This paper refines the Erd ext{"o}s-Niven theorem by characterizing when elementary symmetric functions of reciprocals of arithmetic progression elements are integers, solving an open problem and identifying specific exceptional cases.

Contribution

It provides a complete characterization of when these symmetric functions are integers, extending the classical Erd ext{"o}s-Niven result and resolving an open question from 2012.

Findings

Elementary symmetric functions are not integers except in two specific cases.

The result refines the Erd ext{"o}s-Niven theorem.

Answers an open problem by Chen and Tang (2012).

Abstract

Let $a$ and $b$ be positive integers. In 1946, Erd\H{o}s and Niven proved that there are only finitely many positive integers $n$ for which one or more of the elementary symmetric functions of $1/b, 1/(a+b),..., 1/(an-a+b)$ are integers. In this paper, we show that for any integer $k$ with $1\le k\le n$, the $k$-th elementary symmetric function of $1/b, 1/(a+b),..., 1/(an-a+b)$ is not an integer except that either $b=n=k=1$ and $a\ge 1$, or $a=b=1, n=3$ and $k=2$. This refines the Erd\H{o}s-Niven theorem and answers an open problem raised by Chen and Tang in 2012.