# A Refinement of the $3x+1$ Conjecture

## TL;DR

This paper refines the $3x+1$ conjecture by focusing on numbers congruent to 2 mod 3, revealing new dynamics and characterizations of trajectories, and introducing an accelerated iteration involving specific residue classes.

## Contribution

It introduces a reformulation of the $3x+1$ conjecture focusing on certain residue classes, simplifying the governing function and providing new insights into the problem's dynamics.

## Key findings

- New characterization of $3x+1$ trajectories involving numbers mod 9.
- A simplified governing function with clear partition properties.
- An accelerated iteration focusing on specific residue classes.

## Abstract

We reformulate the $3x+1$ conjecture by restricting attention to numbers congruent to $2$ (mod $3$). This leads to an equivalent conjecture for positive integers that reveals new aspects of the dynamics of the $3x+1$ problem. Advantages include a governing function with particularly simple mapping properties in terms of partitions of the set of integers. We use the refined conjecture to obtain a new characterization of $3x+1$ trajectories that shows a special role played by numbers congruent to $2$ or $8$ (mod $9$). We construct an accelerated iteration whose long-term behavior involves only those numbers.