On the Boundedness of Collatz Sequences

TL;DR

This paper introduces a tree-based approach to the Collatz conjecture, proving the boundedness of Collatz sequences and classifying their behavior into convergence or divergence, while leaving cycle existence unresolved.

Contribution

It presents a novel tree construction for positive odd numbers and proves the boundedness of Collatz sequences, advancing understanding of their long-term behavior.

Findings

Sequences are bounded, ending in 1 or a cycle

All odd numbers are represented in the constructed tree

Cycle existence remains an open question

Abstract

An attempt to come closer to a resolution of the Collatz conjecture is presented. The central idea is the formation of a tree consisting of positive odd numbers with number 1 as root. Functions for generating the tree from the root are presented and paths from nodes to the root are given by modified Collatz sequences (with the even numbers omitted). The Collatz problem is thus reduced to showing that all positive odd numbers are present in the tree. The main result is the proof of the boundedness of Collatz sequences. With the even numbers omitted they either end up in the number 1 (convergence) or in a repetitive cycle of numbers (divergence). The existence/non-existence of cycles in Collatz sequences (with the even numbers omitted) is still an open question.