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The Infinitude of the Primes and Some Coloring Theorems
(Taylor & Francis inc, 2025) Adibelli, Azem Berivan; Goral, Haydar
We first prove the infinitude of the primes via a special case of Rado's theorem whose proof is based on the infinite Ramsey theorem. In the proof, we use the colorings of the positive integers introduced by Levent Alpoge [1] and Andrew Granville [2]. Finally, using Rado's theorem for integral domains, we will give another proof for the infinitude of nonassociated prime elements in any unique factorization domain R with a few units.
A Note on Variants of Euler's Φ-Function
(Univ debrecen, inst Mathematics, 2024) Buyukasik, Engin; Goral, Haydar; Sertbas, Doga Can
It is well-known that the sum of the firstnconsecutive integers alwaysdivides thek-th power sum of the firstnconsecutive integers whenkis odd. Motivatedby this result, in this note, we study the divisibility properties of the power sum ofpositive integers that are coprime tonand not surpassingn. First, we prove a finitenessresult for our divisibility sets using smooth numbers in short intervals. Then, we findthe exact structure of a certain divisibility set that contains the orders of these powersums and our result is of computational flavour.

Scopus İndeksli Yayınlar Koleksiyonu / Scopus Indexed Publications Collection