Cycles in Repeated Exponentiation Modulo $p^n$

TL;DR

This paper investigates the structure and number of cycles in the dynamical system generated by repeated exponentiation modulo prime power $p^n$, providing insights into its periodic behavior.

Contribution

It analyzes the cycle structure of repeated exponentiation modulo $p^n$, a novel focus on prime power moduli in dynamical systems.

Findings

Characterization of cycle counts for $r=p^n$

Identification of conditions for cycle lengths

Quantitative results on the number of cycles

Abstract

Given a number $r$, we consider the dynamical system generated by repeated exponentiations modulo $r$, that is, by the map $u \mapsto f_g(u)$, where $f_g(u) \equiv g^u \pmod r$ and $0 \le f_g(u) \le r-1$. The number of cycles of the defined above dynamical system is considered for $r=p^n$.