A note on the $3x+1$ conjecture

TL;DR

This paper discusses the Collatz conjecture, exploring properties of the map that defines the sequence and providing insights related to the conjecture's validity for all positive integers.

Contribution

It offers new observations and partial results concerning the behavior of the Collatz map, contributing to the understanding of the conjecture.

Findings

Identifies patterns in the iterates of the Collatz map

Provides partial results supporting the conjecture

Suggests potential avenues for further research

Abstract

Let $g$ be a map from the set of positive integers into itself defined as follows: Let $x$ be a positive integer. If $x$ is odd, then $g(x)=3x+1$, and if $x$ is even, then $g(x)=x/2$. The $3x+1$ conjecture, also called the Collatz conjecture, states: For any positive integer $x$ there exists another positive integer $m$ such that the $m$-iterate of $x$ under the map $g$ is equal to $1$, i.e. $g^m(x)=1$. We provide some information related with this conjecture.