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

Abstract

Let $a$ and $b$ be positive integers. In 1946, Erdős 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 Erdős-Niven theorem and answers an open problem raised by Chen and Tang in 2012.

arXiv comment12 pages. To appear in Acta Mathematica Hungarica

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