A simple proof of the Wirsching-Goodwin representation of integers connected to 1 in the $3x+1$ problem

TL;DR

This paper presents a straightforward proof of the Wirsching-Goodwin representation for integers related to 1 in the Collatz conjecture, enabling the computation of all ascending sequences ending at 1 and identifying related periodic sequences.

Contribution

It provides a simple proof of the Wirsching-Goodwin representation, facilitating the analysis of Collatz sequences connected to 1.

Findings

All ascending Collatz sequences ending at 1 can be computed using the representation.

Periodic sequences connected to 1 are identified.

The proof simplifies understanding of the structure of Collatz sequences.

Abstract

This paper gives a simple proof of the Wirsching-Goodwin representation of integers connected to 1 in the $3x+1$ problem (see \cite{Wirsching} and \cite{Goodwin}). This representation permits to compute all the ascending Collatz sequences $(f^{(i)}(n),\: i=1,b-1)$ with a last value $f^{(b)}(n)=1.$ Other periodic sequences connected to $1$ are also identified.