A note on the $3x+1$ conjecture

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.