# The 3n+1 problem: a partition of interest

## TL;DR

This paper explores a novel partitioning of natural numbers related to the Collatz conjecture, proposing a conjugate mapping that suggests all trajectories pass through specific subsets, and discusses special cases where the conjecture might hold.

## Contribution

It introduces a new partitioning approach for the Collatz problem and analyzes special number classes where the conjecture appears to be valid.

## Key findings

- All trajectories pass through certain subsets except the trivial loop.
- The 3n+1 and 3n+3 numbers are unique in allowing such partitions.
- The conjecture holds for these specific classes of numbers.

## Abstract

A mapping conjugate to the Collatz mapping seems to imply that $\N=\{1,2,3,\ldots\}$ is partitioned in a trivial loop $\{1\}$ and `strings' that are ordered subsets of $\{\N \setminus 1\}$ that run from an element of $\{2+3\0\}$ to an element of $\{3+4\0\}$ ($\0=0 \cup \N$). In particular, this means that all trajectories except for the trivial loop go through an element of $\{3+4\0\}$ ($\{5+8\0\}$ for the original mapping). I give reasons for this conjecture. Next, I note that the 3n+1 numbers and the 3n+3 numbers are the only numbers from the generalization $3n+p, p \in \{\ldots,-3,-1,1,3,\ldots\}$ for which such a partition seems to exist. Suspiciously, these are also the only members for which the conjecture (reduction to the trivial loop) seems to hold.