Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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THE DIFFERENCE OF HYPERHARMONIC NUMBERS VIA GEOMETRIC AND ANALYTIC METHODS

  • Altuntas, Cagatay (Department of Mathematics Engineering Faculty of Science and Literature Istanbul Technical University) ;
  • Goral, Haydar (Department of Mathematics Izmir Institute of Technology) ;
  • Sertbas, Doga Can (Department of Mathematics Faculty of Sciences and Literature Cukurova University)
  • Received : 2021.10.12
  • Accepted : 2022.09.01
  • Published : 2022.11.01
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Abstract

Our motivation in this note is to find equal hyperharmonic numbers of different orders. In particular, we deal with the integerness property of the difference of hyperharmonic numbers. Inspired by finiteness results from arithmetic geometry, we see that, under some extra assumption, there are only finitely many pairs of orders for two hyperharmonic numbers of fixed indices to have a certain rational difference. Moreover, using analytic techniques, we get that almost all differences are not integers. On the contrary, we also obtain that there are infinitely many order values where the corresponding differences are integers.

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Acknowledgement

The authors are deeply grateful to the anonymous referee for many constructive remarks and suggestions, especially for pointing out Fact 3.12 which improved the error term in Theorem B.

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