Scopus İndeksli Yayınlar Koleksiyonu / Scopus Indexed Publications Collection
Permanent URI for this collectionhttps://hdl.handle.net/11147/7148
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Article On Classification of Sequences Containing Arbitrarily Long Arithmetic Progressions(World Scientific Publishing, 2023) Cam Çelik, Şermin; Eyidoğan, Sadık; Göral, Haydar; Sertbaş, Doğa CanIn this paper, we study the classification of sequences containing arbitrarily long arithmetic progressions. First, we deal with the question how the polynomial map n(s) can be extended so that it contains arbitrarily long arithmetic progressions. Under some growth conditions, we construct sequences which contain arbitrarily long arithmetic progressions. Also, we give a uniform and explicit arithmetic progression rank bound for a large class of sequences. Consequently, a dichotomy result is deduced on the finiteness of the arithmetic progression rank of certain sequences. Therefore, in this paper, we see a way to determine the finiteness of the arithmetic progression rank of various sequences satisfying some growth conditions.Article Citation - WoS: 2Citation - Scopus: 3The Green-Tao Theorem and the Infinitude of Primes in Domains(Taylor & Francis, 2022) Göral, Haydar; Özcan, Hikmet Burak; Sertbaş, Doğa CanWe 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.