A window to the Convergence of a Collatz Sequence

TL;DR

This paper explores the convergence properties of Collatz sequences, reducing the problem to sequences of numbers divisible by 3, and introduces a reverse sequence with a conjecture about its convergence.

Contribution

It provides an elementary proof of the non-monotonicity of Collatz sequences and defines a reverse sequence with a new conjecture about its convergence to multiples of 3.

Findings

Collatz sequences are not monotonic in their progression.

Convergence of all Collatz sequences can be linked to those divisible by 3.

A new reverse Collatz sequence is introduced with a conjecture on its convergence.

Abstract

In this article, we reduce the unsolved problem of convergence of Collatz sequences to convergence of Collatz sequences of odd numbers that are divisible by 3. We give an elementary proof of the fact that a Collatz sequence does not increase monotonically. We define a unique reverse Collatz sequence and conjecture that this sequence always converges to a multiple of $3$.