preprints.org > computer science and mathematics > mathematics > doi: 10.20944/preprints202408.2050.v1 (registering DOI)

Preprint Article Version 1 This version is not peer-reviewed

Version 1 : Received: 28 August 2024 / Approved: 28 August 2024 / Online: 28 August 2024 (11:44:44 CEST)

Let an odd integer $\mathcal{X}$ be represented as $\sum \limits_{M > m} 2^M + 2^m - 1$ for $m \geq 1$, where $2^m - 1$ is called the Governor. The general dynamics is such that applying the Collatz-type function iteratively lead to $\sum \limits_{N > 1} 2^N + 2^{\mathcal{T}} - 1$, where $2^{\mathcal{T}} - 1$ is referred to as the Trivial Governor. For the $3\mathcal{Z} + 1$ sequence, the Trivial Governor is $2^1 - 1$, while for the $5\mathcal{Z} + 1$ sequence, the Trivial Governors are $2^2 - 1$ and $2^1 - 1$. It is demonstrated that for $\mathcal{X}$ to reappear in a Collatz-type sequence, its Governor must be the Trivial Governor. Specifically, in the $3\mathcal{Z} + 1$ sequence, ancestor mapping shows that odd integers in a repeating cycle are separated by two even integers. Successor mapping further indicates that there are no auxiliary cycles, as the Trivial Governor is transformed into a Governor with a different index. Similarly, in the $5\mathcal{Z} + 1$ sequence, successor mapping reveals that the smallest odd integers that form an auxiliary cycle are found between $2^2$ and $2^5$. Finally, attempts to construct integers that diverge in 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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