THE DIFFERENCE OF HYPERHARMONIC NUMBERS VIA GEOMETRIC AND ANALYTIC METHODS

Altuntaş Ç., Goral H., SERTBAŞ D. C.

JOURNAL OF THE KOREAN MATHEMATICAL SOCIETY, vol.59, no.6, pp.1103-1137, 2022 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 59 Issue: 6
  • Publication Date: 2022
  • Doi Number: 10.4134/jkms.j210630
  • Journal Name: JOURNAL OF THE KOREAN MATHEMATICAL SOCIETY
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH
  • Page Numbers: pp.1103-1137
  • Keywords: Harmonic numbers, arithmetic geometry, prime numbers
  • Istanbul Technical University Affiliated: Yes

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 finite-ness results from arithmetic geometry, we see that, under some extra assumption, there are only finitely many pairs of orders for two hyper -harmonic 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.