The elementary symmetric functions of reciprocal arithmetic progressions
The elementary symmetric functions of reciprocal arithmetic progressions
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发表时间：
2013-11
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通讯作者：
Chunlin Wang;Shaofang Hong
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作者：
Chunlin Wang;Shaofang Hong
设$a$和$b$为正整数。1946年Erd \H{o} s和Niven证明了只有有限多个正整数$n$，其中$1/b, 1/(a+b),..., 1/(an-a+b)$的一个或多个初等对称函数是整数。本文证明了对于任意整数$k$和$1\le k\le n$, $1/b, 1/(a+b),..., 1/(an-a+b)$的$k$ -初等对称函数除了$b=n=k=1$和$a\ge 1$，或者$a=b=1, n=3$和$k=2$之外都不是整数。这加强了Erd \H{o} s-Niven定理，并回答了Chen和Tang在2012年提出的一个开放问题。
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 strengthens the Erd\H{o}s-Niven theorem and answers an open problem raised by Chen and Tang in 2012.