Almost all hyperharmonic numbers are not integers
| Autor: | Haydar Göral, Doğa Can Sertbaş |
|---|---|
| Přispěvatelé: | [Goral, Haydar] Koc Univ, Dept Math, Rumelifeneri Yolu, TR-34450 Istanbul, Turkey -- [Sertbas, Doga Can] Cumhuriyet Univ, Dept Math, Fac Sci, TR-58140 Sivas, Turkey |
| Jazyk: | angličtina |
| Rok vydání: | 2017 |
| Předmět: |
Discrete mathematics
Algebra and Number Theory 010102 general mathematics Order (ring theory) Hyperharmonic number 01 natural sciences 010101 applied mathematics Combinatorics Finite field Integer Harmonic numbers Prime number theory Harmonic number 0101 mathematics Algebraic number Prime power Hyperharmonic numbers Mathematics Prime number theorem |
| Popis: | WOS: 000386418700026 It is an open question asked by Mezo that there is no hyperharmonic integer except 1. So far it has been proved that all hyperharmonic numbers are not integers up to order r = 25. In this paper, we extend the current results for large orders. Our method will be based on three different approaches, namely analytic, combinatorial and algebraic. From analytic point of view, by exploiting primes in short intervals we prove that almost all hyperharmonic numbers are not integers. Then using combinatorial techniques, we show that if n is even or a prime power, or r is odd then the corresponding hyperharmonic number is not integer. Finally as algebraic methods, we relate the integerness property of hyperharmonic numbers with solutions of some polynomials in finite fields. (C) 2016 Elsevier Inc. All rights reserved. Nesin Mathematics Village The authors are very grateful to the Nesin Mathematics Village for their support and warm hospitality during this work. |
| Databáze: | OpenAIRE |
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