Mathematics / Matematik

Permanent URI for this collectionhttps://hdl.handle.net/11147/8

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  • Article
    Citation - WoS: 2
    Citation - Scopus: 1
    The Difference of Hyperharmonic Numbers Via Geometric and Analytic Methods
    (Korean Mathematical Society, 2022) Altuntaş, Çağatay; Göral, Haydar; Sertbaş, Doğa Can; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
    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.
  • Article
    Citation - WoS: 2
    Citation - Scopus: 2
    Rings With Few Units and the Infinitude of Primes
    (Hacettepe Üniversitesi, 2020) Özcan, Hikmet Burak; Taşkın, Sedef; 01. Izmir Institute of Technology
    In this short note, our aim is to provide novel proofs for the infinitude of primes in an algebraic way. It’s thought that the first proof for the infinitude of primes was given by the Ancient Greek mathematician Euclid. To date, most of the proofs have been based on the fact that every positive integer greater than 1 can be written as a product of prime numbers. However, first we are going to prove a ring theoretic fact that if R is an infinite commutative ring with unity and the cardinality of the set of invertible elements is strictly less than the cardinality of the ring, then there are infinitely many maximal ideals. This fact leads to an elegant proof for the infinitude of primes. In addition, under the same cardinality assumption, we consider the special case in which R is a unique factorization domain (for short UFD) and establish another ring theoretic result. Thanks to it, we give a second proof of the infinitude of primes. © 2020, Hacettepe University. All rights reserved.