A Dual-Radix Modular Division Algorithm for Computing Periodic Orbits within Syracuse Dynamical Systems

TL;DR

This paper introduces a dual-radix modular division algorithm to analyze periodic orbits in Syracuse dynamical systems within a new algebraic framework, enabling efficient computation of graded expansions and testing number integrality.

Contribution

The paper presents a novel dual-radix division algorithm tailored for graded $n$-adic integers, advancing the analysis of periodic orbits in Syracuse systems.

Findings

New dual-radix division algorithm for graded $n$-adic integers.

Two methods for testing B"{o}hm-Sontacchi number integrality.

Enhanced understanding of periodic orbits in Syracuse dynamical systems.

Abstract

This article analyzes the periodic orbits of Syracuse dynamical systems in a novel algebraic setting: the commutative ring of graded $n$-adic integers. Within this context, this article introduces a dual-radix modular division algorithm for computing the graded canonical expansions and graded quotients for a certain class of rational expressions that arise from periodic orbits within these dynamical systems. This division algorithm yields two novel methods for testing the integrality of the B\"{o}hm-Sontacchi numbers.