Integer Representations and Trajectories of the 3x+1 Problem

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.