Cellular Automata to More Efficiently Compute the Collatz Map

TL;DR

This paper introduces three cellular automata models that efficiently simulate the Collatz map in bases 2, 3, and 4, enabling faster and parallel computation of Collatz trajectories, which could aid in understanding the conjecture.

Contribution

The paper presents novel cellular automata that transform the Collatz problem into local digit transformations, bypassing certain calculations and enabling parallel processing.

Findings

Binary CA bypasses division by two

Multiple trajectories computed simultaneously

Significant speedup over sequential methods

Abstract

The Collatz, or 3x+1, Conjecture claims that for every positive integer n, there exists some k such that T^k(n)=1, where T is the Collatz map. We present three cellular automata (CA) that transform the global problem of mimicking the Collatz map in bases 2, 3, and 4 into a local one of transforming the digits of iterates. The CAs streamline computation first by bypassing calculation of certain parts of trajectories: the binary CA bypasses division by two altogether. In addition, they allow for multiple trajectories to be calculated simultaneously, representing both a significant improvement upon existing sequential methods of computing the Collatz map and a demonstration of the efficacy of using a massively parallel approach with cellular automata to tackle iterative problems like the Collatz Conjecture.