The Difference of Hyperharmonic Numbers via Geometric and Analytic Methods

\c{C}a\u{g}atay Altunta\c{s}

doi:10.4134/JKMS.j210630

Journal of the
Korean Mathematical Society JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2022; 59(6): 1103-1137

Online first article October 26, 2022 Printed November 1, 2022

https://doi.org/10.4134/JKMS.j210630

The Difference of Hyperharmonic Numbers via Geometric and Analytic Methods

\c{C}a\u{g}atay Altunta\c{s} , Haydar G\"{o}ral, Do\u{g}a Can Sertba\c{s}

Istanbul Technical University; Izmir Institute of Technology; \c{C}ukurova University

Abstract

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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.

Keywords: Harmonic numbers, arithmetic geometry, prime numbers

MSC numbers: 11B83, 11D41, 11N05