A Dual-Radix Approach to Steiner's 1-Cycle Theorem

TL;DR

This paper provides algebraic proofs of Steiner's 1-Cycle Theorem and shows that, under certain bounds, only specific 1-cycles exist in the 3x-1 system, enhancing understanding of its dynamics.

Contribution

It introduces multiple algebraic proofs of Steiner's 1-Cycle Theorem and characterizes the only 1-cycles under exponential bounds in the 3x-1 system.

Findings

Algebraic proofs of Steiner's 1-Cycle Theorem

Identification of only (1) and (5,7) as 1-cycles under bounds

Enhanced understanding of 3x-1 system dynamics

Abstract

This article presents a variety of algebraic proofs of Steiner's $1$-Cycle Theorem. It also demonstrates that, under an exponential upper-bound on the iterates, the only $1$-cycles in the (accelerated) $3x-1$ dynamical system are $(1)$ and $(5,7)$.