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

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.