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Crafting Integers That ‘May’ Diverge for Collatz Sequence
Version 1
: Received: 25 July 2024 / Approved: 26 July 2024 / Online: 26 July 2024 (13:38:29 CEST)
How to cite: Goyal, G. Crafting Integers That ‘May’ Diverge for Collatz Sequence. Preprints 2024, 2024072192. https://doi.org/10.20944/preprints202407.2192.v1 Goyal, G. Crafting Integers That ‘May’ Diverge for Collatz Sequence. Preprints 2024, 2024072192. https://doi.org/10.20944/preprints202407.2192.v1
Abstract
Let an odd integer be represented as $\sum_{n > m} 2^n + 2^m - 1$ for $m \geq 1$. To transform $2^m - 1$ according to the Collatz rules, the $3x + 1$ step is applied, resulting in the even integer $2^{m + 1} + 2^m - 2$. This even integer, being exactly once divisible by 2, becomes the next odd integer $2^m + 2^{m - 1} - 1$. This represents the growth phase, as exactly one even step follows an odd step. However, while the overall value of the integer increases, terms with decreasing indices are generated after each odd-even cycle. After $m$ such cycles, a term $2^{m - m}$ (which equals 1) is generated, canceling the negative 1. Additional even steps are then performed until the integer becomes odd again, which occurs when the next smallest index becomes zero. The positive 1 thus obtained is written as $2^1 - 1$. If uninterrupted, the term $2^1 - 1$ has the trivial cycle $2^1 - 1 \rightarrow 2^2 \rightarrow 2^1 \rightarrow 2^1 - 1$. This represents the shrinkage phase, as two even steps follow an odd step. To interrupt the trivial cycle, a sequence such as $2^1 + 2^2 + \cdots$ must be available to combine with $2^1 - 1$ and produce $2^{\mathcal{M}} - 1$. This starts another growth phase that lasts for $\mathcal{M}$ odd-even cycles. Finally, there is an attempt to craft an integer that can escape the shrinkage phase in the Collatz dynamics.
Keywords
collatz; 3x+1
Subject
Computer Science and Mathematics, Mathematics
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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