# Integer Representations and Trajectories of the 3x+1 Problem

## TL;DR

This paper explores the trajectories of the Collatz function, proposing a new representation for integers and showing that proving a specific relation for odd numbers could resolve the Collatz Conjecture.

## Contribution

It introduces a novel integer representation linked to Collatz trajectories and reduces the conjecture to verifying a relation for odd numbers.

## Key findings

- Derived a new integer representation involving powers of 2 and 3.
- Showed that proving a specific relation for odd numbers suffices to prove the Collatz Conjecture.
- Connected the Collatz problem to a new algebraic condition on odd integers.

## Abstract

This paper studies certain trajectories of the Collatz function. I show that if for each odd number $n$, $n\sim 3n+2$ then every positive integer $n \in \mathbb{N}\setminus 2^{\mathbb{N}}$ has the representation $$n=\left(2^{a_{k+1}}-\sum_{i=0}^{k}{2^{a_i}3^{k-i}}\right)/ 3^{k+1}$$ where $0\le a_0 \le a_1 \le \cdot \cdot \cdot \le a_{k+1}$. As a consequence, in order to prove Collatz Conjecture I illustrate that it is sufficient to prove $n\sim 3n+2$ for any odd $n\in \mathbb{N}\setminus 2^{\mathbb{N}} $. This is the main result of the paper.