Scopus İndeksli Yayınlar Koleksiyonu / Scopus Indexed Publications Collection

Permanent URI for this collectionhttps://hdl.handle.net/11147/7148

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Institution Author: search.filters.institutionAuthor.Özcan, Hikmet Burak

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  • Article
    Citation - WoS: 2
    Citation - Scopus: 3
    The Green-Tao Theorem and the Infinitude of Primes in Domains
    (Taylor & Francis, 2022) Göral, Haydar; Göral, Haydar; Özcan, Hikmet Burak; Sertbaş, Doğa Can; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
    We first prove an elementary analogue of the Green-Tao Theorem. The celebrated Green-Tao Theorem states that there are arbitrarily long arithmetic progressions in the set of prime numbers. In fact, we show the Green-Tao Theorem for polynomial rings over integral domains with several variables. Using the Generalized Polynomial van der Waerden Theorem, we also prove that in an infinite unique factorization domain, if the cardinality of the set of units is strictly less than that of the domain, then there are infinitely many prime elements. Moreover, we deduce the infinitude of prime numbers in the positive integers using polynomial progressions of length three. In addition, using unit equations, we provide two more proofs of the infinitude of prime numbers. Finally, we give a new proof of the divergence of the sum of reciprocals of all prime numbers.
  • 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.