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Multiple reciprocal sums and multiple reciprocal star sums of polynomials are almost never integers

Research paper by Shaofang Hong, Liping Yang, Qiuyu Yin, Min Qiu

Indexed on: 21 Mar '17Published on: 21 Mar '17Published in: arXiv - Mathematics - Number Theory



Abstract

Let $n$ and $k$ be integers such that $1\le k\le n$ and $f(x)$ be a nonzero polynomial of nonnegative integer coefficients. For any $n$-tuple $s=(s_1, ..., s_n)$ of positive integers, we define $$H_{k,f}(s, n):=\sum\limits_{1\leq i_{1}<\cdots<i_{k}\le n} \prod\limits_{j=1}^{k}\frac{1}{f(i_{j})^{s_j}}$$ and $$H_{k,f}^*(s, n):=\sum\limits_{1\leq i_{1}\leq \cdots\leq i_{k}\leq n} \prod\limits_{j=1}^{k}\frac{1}{f(i_{j})^{s_j}}.$$ If all $s_j$ are 1, then let $H_{k,f}(s, n):=H_{k,f}(n)$ and $H_{k,f}^*(s, n):=H_{k,f}^*(n)$. Hong and Wang refined the results of Erd\"{o}s and Niven, and of Chen and Tang by showing that $H_{k,f}(n)$ is not an integer if $n\geq 4$ and $f(x)=ax+b$ with $a$ and $b$ being positive integers. Meanwhile, Luo, Hong, Qian and Wang proved that the similar result holds if $f(x)$ is of nonnegative integer coefficients and of degree no less than two. KH. Hessami Pilehrood, T. Hessami Pilehrood and Tauraso showed that if $f(x)=x$, then $H_{k,f}(s,n)$ and $H_{k,f}^*(s,n)$ are nearly never integers. In this paper, we show that if either $\deg f(x)\ge 2$ or $f(x)$ is linear and $s_j\ge 2$ for all integers $j$ with $1\le j\le n$, then $H_{k,f}(s, n)$ and $H_{k,f}^*(s, n)$ are not integers except for the case $f(x)=x^{m}$ with $m\geq1$ being an integer and $n=k=1$, in which case, both of $H_{k,f}(s, n)$ and $H_{k,f}^*(s, n)$ are integers.