关于$1,1/3^{s_{1}},...,1/(2n-1)^{s_{n-1}}$的第二类初等对称函数的整性

设$d,\ m$ 与 $n$ 均为正整数. 在1915年, Theisinger证明当$n\ge 2$时,$n$次调和和 $\sum_{k=1}^n\frac{1}{k}$不是一个整数. 在1946年,Erd\H{o}s和Niven 证明仅有有限多个$n$, 使得关于$1/m, 1/(m+d),..., 1/(m+nd)$ 的一个或多个初等对称函数是整数.在2015年, Wang 和 Hong 证明当 $n\ge 2$ 时,$1,1/3,...,1/(2n-1)$ 的所有初等对称函数均非整数.在本文中, 我们证明如下结果成立: 如果$n\ge 2$为正整数, 那么对任意$n$个正整数 $s_0,..., s_{n-1}$, 关于$1,1/3^{s_{1}},...,1/(2n-1)^{s_{n-1}}$的第二类初等对称函数 $$\sum\limits_{0\le i

On the integrality of the second elementary symmetric function of $1,1/3^{s_{1}},...,1/(2n-1)^{s_{n-1}}

Let $d,m$ and $n$ be positive integers. In 1915, Theisinger proved that if $n\ge 2$, then the $n$-th harmonic sum $\sum_{k=1}^n\frac{1}{k}$ is not an integer. In 1946, Erd\H{o}s and Niven extended Theisinger''s theorem by showing that 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. In 2015, Wang and Hong proved that none of the elementary symmetric functions of $1,1/3,...,1/(2n-1)$ is an integer if $n\ge 2$. In this paper, we show that if $n\ge 2$, then for arbitrary $n$ positive integers $s_0, ..., s_{n-1}$ (not necessarily distinct and not necessarily monotonic),the following multiple reciprocal power sum $$\sum\limits_{0\le i