Volume 35, Issue 1 (2026)

A Small Collatz Rule without the Plus One

Kevin Knight
Threeven Labs
134 Roma Court
Marina del Rey, CA, 90292, USA

Abstract

The Collatz rule is one of the earliest examples of a simple, deterministic system that produces chaotic behavior. The rule takes any odd positive integer n to 3n+1 and any even positive integer n to n/2. Iterating this rule yields complex sequences whose dynamics are poorly understood; for example, it is unknown whether all such sequences reach 1 (the Collatz conjecture). It is reasonable to suspect that this complexity derives from the interplay of multiplication (3n) and addition (+1). However, in 2002, Monks was able to drop the +1 by constructing a 1$$021$$020-condition rule that simulates the Collatz rule using only multiplication. Monks’s rule greatly simplifies the Collatz dynamics but at the cost of an enormous rule. The current paper achieves the goal of removing addition with a significantly smaller 30-condition rule. We show how this rule replicates the Collatz process, and we place conditions on any cyclic trajectory purporting to be a counterexample to the Collatz conjecture.

Keywords: number theory; dynamical systems

Cite this publication as:
K. Knight, “A Small Collatz Rule without the Plus One,” Complex Systems, 35(1), 2026 pp. 1–9.
https://doi.org/10.25088/ComplexSystems.35.1.1