Acta Mathematica Hungarica

, Volume 157, Issue 2, pp 522–536 | Cite as

A generalization of a theorem of Nagell

  • Y. L. Feng
  • S. F. HongEmail author
  • X. Jiang
  • Q. Y. Yin


Let n be a positive integer. Theisinger [7] proved that if \({n\ge 2}\) , then the n-th harmonic sum \({\sum_{k=1}^n\frac{1}{k}}\) is not an integer. Let a and b be positive integers. Nagell [6] extended Theisinger’s theorem by showing that the reciprocal sum \({\sum_{k=1}^{n}\frac{1}{a+(k-1)b}}\) is not an integer if \({n\ge 2}\) . Erdős and Niven [2] proved a theorem of a similar nature that states that there is only a finite number of integers n for which one or more of the elementary symmetric functions of \({1, 1/2, \ldots, 1/n}\) is an integer. We present a generalization of Nagell’s theorem. In fact, we show that for arbitrary n positive integers \({s_1, \ldots, s_n}\) (not necessarily distinct and not necessarily monotonic), the reciprocal power sum
is never an integer if \({n\ge 2}\) . The proof of our result is analytic and p-adic in character.

Key words and phrases

p-adic valuation arithmetic progression reciprocal power sum Bertrand’s postulate integrality 

Mathematics Subject Classification

primary 11N13 11B25 11B83 11B75 


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The authors thank the anonymous referee for careful reading of the manuscript and helpful comments.


  1. 1.
    Chen Y.G., Tang M.: On the elementary symmetric functions of \({1, 1/2, \ldots, 1/n}\) . Amer. Math. Monthly. 119, 862–867 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Erdős P., Niven I.: Some properties of partial sums of the harmonic series. Bull. Amer. Math. Soc., 52, 248–251 (1946)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Hong S.F., Wang C.L.: The elementary symmetric functions of reciprocals of the elements of arithmetic progressions. Acta Math. Hungar., 144, 196–211 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    N. Koblitz, p-adic Numbers, p-adic Analysis and Zeta-functions, 2nd ed., Springer-Verlag (New York, 1984).Google Scholar
  5. 5.
    Luo Y.Y., Hong S.F., Qian G.Y., Wang C.L.: The elementary symmetric functions of a reciprocal polynomial sequence. C. R. Acad. Sci. Paris, Ser. I, 352, 269–272 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Nagell T.: Eine Eigenschaft gewissen Summen. Skr. Norske Vid. Akad. Kristiania, 13, 10–15 (1923)zbMATHGoogle Scholar
  7. 7.
    Theisinger L.: Bemerkung über die harmonische Reihe, Monatsh. Math. Phys., 26, 132–134 (1915)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Wang C.L., Hong S.F.: On the integrality of the elementary symmetric functions of \({1, 1/3, \ldots, 1/(2n-1)}\) . Math. Slovaca, 65, 957–962 (2015)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Yang W.X., Li M., Feng Y.L., Jiang X.: On the integrality of the first and second elementary symmetric functions of \({1, 1/2^{s_2}, \ldots, 1/n^{s_n}}\) .. AIMS Math., 2, 682–691 (2017)CrossRefGoogle Scholar
  10. 10.
    Yin Q.Y., Hong S.F., Yang L.P., Qiu M.: Multiple reciprocal sums and multiple reciprocal star sums of polynomials are almost never integers. J. Number Theory, 195, 269–292 (2019)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2019

Authors and Affiliations

  • Y. L. Feng
    • 1
  • S. F. Hong
    • 1
    Email author
  • X. Jiang
    • 1
  • Q. Y. Yin
    • 1
  1. 1.Mathematical CollegeSichuan UniversityChengduP. R. China

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