preprints.org > computer science and mathematics > mathematics > doi: 10.20944/preprints202408.2050.v3 (registering DOI)

Preprint Article Version 3 This version is not peer-reviewed

Version 1 : Received: 28 August 2024 / Approved: 28 August 2024 / Online: 28 August 2024 (11:44:44 CEST)
Version 2 : Received: 29 August 2024 / Approved: 29 August 2024 / Online: 29 August 2024 (10:16:53 CEST)
Version 3 : Received: 30 August 2024 / Approved: 30 August 2024 / Online: 30 August 2024 (19:22:09 CEST)

Let an odd integer $\mathcal{X}$ be expressed as $\sum \limits_{M > m} 2^M + 2^m - 1$, where $2^m - 1$ is called the Governor. The general dynamics of the Collatz-type functions is such that a high index Governor is converted to $2^1 - 1$. For the $3\mathcal{Z} + 1$ sequence, the Governor occurring in the trivial cycle is $2^1 - 1$, while for the $5\mathcal{Z} + 1$ sequence, the Trivial Governors are $2^2 - 1$ and $2^1 - 1$. Therefore, for the specific cases of $3\mathcal{Z} + 1$ and $5\mathcal{Z} + 1$ sequences, it can be said that the Collatz function reduces $\mathcal{X}$ to $\sum \limits_{N > \mathcal{T}} 2^N + 2^{\mathcal{T}} - 1$, where $2^{\mathcal{T}} - 1$ is referred to as the Trivial Governor. Further, once the Trivial Governor is reached, it can again evolve to a higher index Governor by interacting with other terms. This feature allows the $\mathcal{X}$ to reappear in a Collatz-type sequence if its original Governor is expressible as a sum of series including the Trivial Governor, that is, $2^m - 1 = \cdots + 2^{\mathcal{T} + 1} + 2^{\mathcal{T}} + 2^{\mathcal{T}} - 1$. Therefore, if $\mathcal{X}$ reappears in the sequence, at least one odd ancestor of $\mathcal{X}$ in the sum of series format must have the Governor $2^m - 1$. Ancestor mapping shows that all the odd ancestors of $\mathcal{X}$ have the Trivial Governor of the respective Collatz sequence implying that $\mathcal{X}$ can repeat only if its original Governor is the Trivial Governor. Specifically for the $3\mathcal{Z} + 1$ sequence, the odd ancestors are separated by two even integers. Successor mapping for the $3\mathcal{Z} + 1$ further indicates that there are no auxiliary cycles, as the Trivial Governor is transformed to a different index Governor. Similarly, successor mapping for the $5\mathcal{Z} + 1$ sequence reveals that the smallest odd integers that form an auxiliary cycle are smaller than $2^5$. Finally, attempts to construct integers that diverge for the $3\mathcal{Z} + 1$ sequence suggest that no such integers exist.

Collatz, 3x+1, 5x+1, collatz conjecture

Computer Science and Mathematics, Mathematics

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