On the Structure and the Behavior of Collatz 3n + 1 Sequences

TL;DR

This paper analyzes the structure of Collatz sequences, showing they consist of finite subsequences with specific properties, and proves that Collatz's conjecture holds if all sequences starting with certain residues have finite stopping times.

Contribution

It introduces a structured decomposition of Collatz sequences and proves the conjecture for sequences with specific residue classes, linking sequence behavior to Fibonacci numbers.

Findings

Sequences are composed of finite structured subsequences.

Collatz conjecture is true if sequences with certain residues have finite stopping times.

Almost all sequences with specific residues have finite stopping times, with statistical evidence supporting this.

Abstract

It is shown that every Collatz sequence $C(s)$ consists only of same structured finite subsequences $C^h(s)$ for $s\equiv9\ (mod\ 12)$ or $C^t(s)$ for $s\equiv3,7\ (mod\ 12)$. For starting numbers of specific residue classes ($mod\ 12\cdot2^h$) or ($mod\ 12\cdot2^{t+1}$) the finite subsequences have the same length $h,t$. It is conjectured that for each $h,t\geq2$ the number of all admissible residue classes is given exactly by the Fibonacci sequence. This has been proved for $2\leq h,t\leq50$. Collatz's conjecture is equivalent to the conjecture that for each $s\in\mathbb{N},s>1$, there exists $k\in\mathbb{N}$ such that $T^k(s)<s$. The least $k\in\mathbb{N}$ such that $T^k(s)<s$ is called the stopping time of $s$, which we will denote by $\sigma(s)$. It is shown that Collatz's conjecture is true, if every starting number $s\equiv3,7\ (mod\ 12)$ have finite stopping time. We denote $\tau(s)$ as the number of $C^t(s)$ until $\sigma(s)$ is reached for a starting number $s\equiv3,7\ (mod\ 12)$. Starting numbers of specific residue classes ($mod\ 3\cdot2^{\sigma(s)}$) have the same stopping times $\sigma(s)$ and $\tau(s)$. By using $\tau(s)$ it is shown that almost all $s\equiv3,7\ (mod\ 12)$ have finite stopping time and statistically two out of three $s\equiv3,7\ (mod\ 12)$ have $\tau(s)=1$. Further it is shown what consequences it entails, if a $C(s)$ grows to infinity.