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http://dx.doi.org/10.4134/JKMS.j210630

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)
Publication Information
Journal of the Korean Mathematical Society / v.59, no.6, 2022 , pp. 1103-1137 More about this Journal
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.
Keywords
Harmonic numbers; arithmetic geometry; prime numbers;
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