The intricate labyrinth of Collatz sequences

TL;DR

This paper refines the understanding of Collatz sequences by narrowing the focus to specific residue classes and demonstrating that sequences either reach a multiple of 3 or are connected through a network of sequences, advancing the analysis of their convergence.

Contribution

It further reduces the subset of odd numbers needed to analyze for Collatz convergence and constructs a network linking all odd sequences, providing new structural insights.

Findings

Sequences of odd numbers divisible by 3 are central to convergence.

Either a Collatz sequence or an equivalent one reaches a multiple of 3.

A network of all odd Collatz sequences is constructed.

Abstract

In a previous article, we reduced the unsolved problem of the convergence of Collatz sequences, to convergence of Collatz sequences of odd numbers, that are divisible by 3. In this article, we further reduce this set to odd numbers that are congruent to 21 mod 24. We also show that either the Collatz sequence of a given odd number or an equivalent Collatz sequence reverses to a multiple of 3. Moreover, we construct a network composed of Collatz sequences of all odd numbers.