A Right Inverse Function for Collatz Function

Abstract

A right inverse function $S(x)$ from the set of natural numbers $N $ into itself for classical Collatz function is defined by \begin{eqnarray*} S(x)=\begin{cases} \text{ $2m+1$}, & \text{if $x=3m+2$} \\ \text{ $6m$}, & \text{if $x=3m$}\\ \text{ $6m+2$}, & \text{if $x=3m+1$.} \end{cases} \end{eqnarray*} This study assumes that the positive integers $x$, satisfying $S^{k}(x)>x$, for some $k$ as integers favourable to Collatz conjecture. The integers of the form $x=3m+2$ poses difficulty for Collatz conjecture. Hence $ N_{3}=\{3m+2:m \in N\}$ is spilitted into many sets till it happens that $y=3^{k+1}p+x$ with $x\in N_{3}$ are favourable to Collatz conjecture (with $p=0, 1, 2,\dots)$.