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AIMS Mathematics

2017, Volume 2, Issue 4: 682-691. doi: 10.3934/Math.2017.4.682
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Research article

On the integrality of the first and second elementary symmetricfunctions of $1, 1/2^{s_2}, ...,1/n^{s_n}$

  • 1.
    Mathematical College, Sichuan University, Chengdu 610064, P.R. China
  • 2.
    School of Mathematics and Statistics, Southwest University, Chongqing 400715, P.R. China
  • Received: 29 September 2017 Accepted: 30 November 2017 Published: 08 December 2017

  • MSC : 11B83, 11B75

  • It is well known that the harmonic sum $H_{n}(1)=\sum_{1\leq k\leq n}\frac{1}{k}$ is never an integer for $n>1$. Erdös and Niven proved in 1946 that the multiple harmonic sum $H_{n}(\{1\}^r)=\sum_{1\leq k_{1} < \cdots < k_{r}\leq n}\frac{1}{k_{1}\cdots k_{r}}$ can take integer values for at most finite many integers $n$. In 2012, Chen and Tang refined this result by showing that $H_{n}(\{1\}^r)$ is an integer only for $(n, r)=(1, 1)$ and $(n, r)=(3, 2)$. In this paper, we consider the integrality problem for the first and second elementary symmetric function of $1, 1/2^{s_2}, ..., $ $1/n^{s_n}$, we show that none of them is an integer with some natural exceptions.

    Citation: Wanxi Yang, Mao Li, Yulu Feng, Xiao Jiang. On the integrality of the first and second elementary symmetricfunctions of $1, 1/2^{s_2}, ...,1/n^{s_n}$[J]. AIMS Mathematics, 2017, 2(4): 682-691. doi: 10.3934/Math.2017.4.682

    Related Papers:

  • It is well known that the harmonic sum $H_{n}(1)=\sum_{1\leq k\leq n}\frac{1}{k}$ is never an integer for $n>1$. Erdös and Niven proved in 1946 that the multiple harmonic sum $H_{n}(\{1\}^r)=\sum_{1\leq k_{1} < \cdots < k_{r}\leq n}\frac{1}{k_{1}\cdots k_{r}}$ can take integer values for at most finite many integers $n$. In 2012, Chen and Tang refined this result by showing that $H_{n}(\{1\}^r)$ is an integer only for $(n, r)=(1, 1)$ and $(n, r)=(3, 2)$. In this paper, we consider the integrality problem for the first and second elementary symmetric function of $1, 1/2^{s_2}, ..., $ $1/n^{s_n}$, we show that none of them is an integer with some natural exceptions.


    加载中
    [1] Y.G. Chen and M. Tang, On the elementary symmetric functions of 1, 1/2, ..., 1/n, Am. Math. Mon., 119 (2012), 862-867.
    [2] P. Erdös and I. Niven, Some properties of partial sums of the harmonic series, B. Am. Math. Soc., 52 (1946), 248-251.
    [3] S.F. Hong and C.L. Wang, The elementary symmetric functions of reciprocals of the elements of arithmetic progressions, Acta Math. Hung., 144 (2014), 196-211.
    [4] N. Koblitz, p-Adic numbers, p-adic analysis and zeta-functions, GTM 58, Springer-Verlag, New York, 1984.
    [5] Y.Y. Luo, S.F. Hong, G.Y. Qian and C.L.Wang, The elementary symmetric functions of a reciprocal polynomial sequence, C.R. Math., 352 (2014), 269-272.
    [6] M. B. Nathanson, Elementary methods in number theroy, GTM 195, Springer-Verlag, New York, 2000.
    [7] C.L. Wang and S.F. Hong, On the integrality of the elementary symmetric functions of 1, 1/3, ..., 1=(2n -1), Math. Slovaca, 65 (2015), 957-962.
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  • © 2017 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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