Let α = α1 α2 ∈ Q \ {0} ; a positive integer N is said to be an α-Korselt number (Kα-number, for short) if N ̸ = α and α2p − α1 divides α2N − α1 for every prime divisor p of N. In this paper we prove that for each squarefree composite number N there exist finitely many rational numbers α such that N is a Kα-number and if α ≤ 1 then N has at least three prime factors. Moreover, we prove that for each α ∈ Q \ {0} there exist only finitely many squarefree composite numbers N with two prime factors such that N is a Kα-number.
Let α ∈ ℤ\{0}. A positive integer N is said to be an α-Korselt number (Kα-number, for short) if N ≠ α and N - α is a multiple of p - α for each prime divisor p of N. By the Korselt set of N, we mean the set of all α ∈ ℤ\{0} such that N is a Kα-number; this set will be denoted by [Formula: see text]. Given a squarefree composite number, it is not easy to provide its Korselt set and Korselt weight both theoretically and computationally. The simplest kind of squarefree composite number is the product of two distinct prime numbers. Even for this kind of numbers, the Korselt set is far from being characterized. Let p, q be two distinct prime numbers. This paper sheds some light on [Formula: see text].
<abstract><p>The concept of spherical hesitant fuzzy set is a mathematical tool that have the ability to easily handle imprecise and uncertain information. The method of aggregation plays a great role in decision-making problems, particularly when there are more conflicting criteria. The purpose of this article is to present novel operational laws based on the Yager t-norm and t-conorm under spherical hesitant fuzzy information. Furthermore, based on the Yager operational laws, we develop the list of Yager weighted averaging and Yager weighted geometric aggregation operators. The basic fundamental properties of the proposed operators are given in detail. We design an algorithm to address the uncertainty and ambiguity information in multi-criteria group decision making (MCGDM) problems. Finally, a numerical example related to Parkinson disease is presented for the proposed model. To show the supremacy of the proposed algorithms, a comparative analysis of the proposed techniques with some existing approaches and with validity test is presented.</p></abstract>
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