A Proof of Collatz Conjecture Based on a New Tree Topology

TL;DR

This paper claims to prove the Collatz conjecture by introducing a novel tree topology that demonstrates all integers eventually reach 1 without cycles, providing a new structural perspective on the problem.

Contribution

It introduces a new tree topology for the Collatz conjecture and proves that all integers are uniquely distributed on this tree, ensuring all trajectories end at 1.

Findings

All integers are uniquely distributed on the proposed tree.

No cycles other than 1-2-1 exist in the tree.

Every trajectory terminates at 1.

Abstract

Consider a finite positive integer. If it is even, divide it by 2, and if it is odd, multiply it by 3 and add 1. This will give you a new integer. Following the procedure for the new integer, you will receive another integer. Repeat the steps, and after a few repetitions, you will finally reach 1. Collatz conjecture states that the final integer in the mentioned process will always be 1, no matter what integer it starts with. Although the procedure of the conjecture is easy to describe, its correctness has not yet been confirmed. This article proves the conjecture by introducing a tree topology of it. Given the proposed tree, we can prove that all integers are uniquely distributed on the tree, and there is no cycle other than 1-2-1. We also see how every trajectory ends in 1.