Generalization of Some Arithmetical Properties of Fermat-Euler Dynamical Systems

TL;DR

This paper generalizes and extends the arithmetical properties of Fermat-Euler dynamical systems classes introduced by Arnold, revealing that Arnold's specific cases are special instances of broader properties.

Contribution

It introduces a general framework for classes (2^k+) and (2^k-) and demonstrates that Arnold's earlier results are specific cases within this broader theory.

Findings

Established general properties of classes (2^k+) and (2^k-)

Showed Arnold's properties are special cases of the new framework

Extended understanding of Fermat-Euler dynamical systems

Abstract

We study and generalize some arithmetical properties of the classes (2^k+) and (2^k-) introduced by V. I. Arnold: a number n belongs to the class (N+) if N|\varphi(n) and 2^{\frac{\varphi(n)}{N}} \equiv 1 mod n where \varphi(n) is the Euler function, and belongs to the class (M-) if M|\varphi(n) and 2^{\frac{\varphi(n)}{M}} \equiv -1 mod n. The classes (2+), (2-),(4+), (4-), (8+)and (8-) are studied by V. I. Arnold and here we will show general properties of the classes (2^k+) and (2^k-) and we will see that the properties which is proved by V. I. Arnold are special cases of ours.