Search Advanced

AIMS Mathematics

2017, Volume 2, Issue 4: 682-691. doi: 10.3934/Math.2017.4.682
Previous Article Next Article
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.
  • Reader Comments
  • © 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)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
AIMS Mathematics
1.8 3.4

Metrics

Article views(4161) PDF downloads(1022) Cited by(3)

Preview PDF
Download XML
Export Citation
Article outline

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog