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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
Article

Abstract

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
$$\sum_{k=1}^{n}\frac{1}{(a+(k-1)b)^{s_{k}}}$$
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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Notes

Acknowledgment

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

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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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