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THE DIFFERENCE OF HYPERHARMONIC NUMBERS VIA GEOMETRIC AND ANALYTIC METHODS
Altuntas, Cagatay,Goral, Haydar,Sertbas, Doga Can Korean Mathematical Society 2022 대한수학회지 Vol.59 No.6
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.
ON THE p-ADIC VALUATION OF GENERALIZED HARMONIC NUMBERS
Cagatay Altuntas 대한수학회 2023 대한수학회보 Vol.60 No.4
For any prime number $p$, let $J(p)$ be the set of positive integers $n$ such that the numerator of the $n^{th}$ harmonic number in the lowest terms is divisible by this prime number $p$. We consider an extension of this set to the generalized harmonic numbers, which are a natural extension of the harmonic numbers. Then, we present an upper bound for the number of elements in this set. Moreover, we state an explicit condition to show the finiteness of our set, together with relations to Bernoulli and Euler numbers.