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- W1654407577 abstract "It is shown that every Collatz sequence $C(s)$ consists only of same structured finite subsequences $C^h(s)$ for $sequiv9 (mod 12)$ or $C^t(s)$ for $sequiv3,7 (mod 12)$. For starting numbers of specific residue classes ($mod 12cdot2^h$) or ($mod 12cdot2^{t+1}$) the finite subsequences have the same length $h,t$. It is conjectured that for each $h,tgeq2$ the number of all admissible residue classes is given exactly by the Fibonacci sequence. This has been proved for $2leq h,tleq50$. Collatz's conjecture is equivalent to the conjecture that for each $sinmathbb{N},s>1$, there exists $kinmathbb{N}$ such that $T^k(s)<s$. The least $kinmathbb{N}$ 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 $sequiv3,7 (mod 12)$ have finite stopping time. We denote $tau(s)$ as the number of $C^t(s)$ until $sigma(s)$ is reached for a starting number $sequiv3,7 (mod 12)$. Starting numbers of specific residue classes ($mod 3cdot2^{sigma(s)}$) have the same stopping times $sigma(s)$ and $tau(s)$. By using $tau(s)$ it is shown that almost all $sequiv3,7 (mod 12)$ have finite stopping time and statistically two out of three $sequiv3,7 (mod 12)$ have $tau(s)=1$. Further it is shown what consequences it entails, if a $C(s)$ grows to infinity." @default.
- W1654407577 created "2016-06-24" @default.
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- W1654407577 date "2014-11-27" @default.
- W1654407577 modified "2023-09-27" @default.
- W1654407577 title "On the structure and the behaviour of Collatz 3n + 1 sequences - Finite subsequences and the role of the Fibonacci sequence" @default.
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