Mathematics / Matematik
Permanent URI for this collectionhttps://hdl.handle.net/11147/8
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Article Arithmetic Progressions in Certain Subsets of Finite Fields(Elsevier, 2023) Eyidoğan, Sadık; Göral, Haydar; Kutlu, Mustafa KutayIn this note, we focus on how many arithmetic progressions we have in certain subsets of finite fields. For this purpose, we consider the sets Sp = {t2 : t & ISIN; Fp} and Cp = {t3 : t & ISIN; Fp}, and we use the results on Gauss and Kummer sums. We prove that for any integer k & GE; 3 and for an odd prime number p, the number of k-term arithmetic progressions in Sp is given by p2 2k + R, where and ck is a computable constant depending only on k. The proof also uses finite Fourier analysis and certain types of Weil estimates. Also, we obtain some formulas that give the exact number of arithmetic progressions of length in the set Sp when & ISIN; {3,4, 5} and p is an odd prime number. For = 4, 5, our formulas are based on the number of points onArticle 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: 1The Difference of Hyperharmonic Numbers Via Geometric and Analytic Methods(Korean Mathematical Society, 2022) Altuntaş, Çağatay; Göral, Haydar; Sertbaş, Doğa CanOur 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 Lehmer’s Conjecture Via Model Theory(Japan Academy, 2022) Göral, HaydarIn this short note, we study Lehmer's conjecture in terms of stability theory. We state Bounded Lehmer's conjecture, and we prove that if a certain formula is uniformly stable in a class of structures, then Bounded Lehmer's conjecture holds. Our proof is based on Van der Waerden's theorem from additive combinatoricsArticle Citation - WoS: 1Citation - Scopus: 1Applications of Class Numbers and Bernoulli Numbers To Harmonic Type Sums(Korean Mathematical Society, 2021) Göral, Haydar; Sertbaş, Doğa CanDivisibility properties of harmonic numbers by a prime number p have been a recurrent topic. However, finding the exact p-adic orders of them is not easy. Using class numbers of number fields and Bernoulli numbers, we compute the exact p-adic orders of harmonic type sums. Moreover, we obtain an asymptotic formula for generalized harmonic numbers whose p-adic orders are exactly one.Article A Note on Points on Algebraic Sets(Hacettepe Üniversitesi, 2021) Çam Çelik, Şermin; Göral, HaydarIn this short note, we count the points on algebraic sets which lie in a subset of a domain. It is proved that the set of points on algebraic sets coming from certain subsets of a domain has the full asymptotic. This generalizes the first theorem of [E. Alkan and E.S. Yoruk, Statistics and characterization of matrices by determinant and trace, Ramanujan J., 2019] and also anwers some questions from the same article.