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Version 5
This version is not peer-reviewed
$2^m - 1$ as the Integer Formulation to Govern the Dynamics of Collatz-Type Sequences
Version 1
: Received: 10 July 2024 / Approved: 11 July 2024 / Online: 11 July 2024 (12:28:15 CEST)
Version 2 : Received: 17 July 2024 / Approved: 18 July 2024 / Online: 18 July 2024 (14:33:11 CEST)
Version 3 : Received: 19 July 2024 / Approved: 22 July 2024 / Online: 22 July 2024 (11:55:52 CEST)
Version 4 : Received: 22 July 2024 / Approved: 23 July 2024 / Online: 23 July 2024 (16:06:28 CEST)
Version 5 : Received: 24 July 2024 / Approved: 25 July 2024 / Online: 25 July 2024 (14:59:59 CEST)
Version 2 : Received: 17 July 2024 / Approved: 18 July 2024 / Online: 18 July 2024 (14:33:11 CEST)
Version 3 : Received: 19 July 2024 / Approved: 22 July 2024 / Online: 22 July 2024 (11:55:52 CEST)
Version 4 : Received: 22 July 2024 / Approved: 23 July 2024 / Online: 23 July 2024 (16:06:28 CEST)
Version 5 : Received: 24 July 2024 / Approved: 25 July 2024 / Online: 25 July 2024 (14:59:59 CEST)
How to cite: Goyal, G. $2^m - 1$ as the Integer Formulation to Govern the Dynamics of Collatz-Type Sequences. Preprints 2024, 2024070961. https://doi.org/10.20944/preprints202407.0961.v5 Goyal, G. $2^m - 1$ as the Integer Formulation to Govern the Dynamics of Collatz-Type Sequences. Preprints 2024, 2024070961. https://doi.org/10.20944/preprints202407.0961.v5
Abstract
It has been discovered that representing odd integers as modified binary expressions of the form $\sum \limits_{n > m}2^n + 2^m - 1$ for $m \geq 1$ helps in understanding the dynamics of Collatz-type sequences. Starting with the original Collatz sequence $3x + 1$, it is found that when the odd step is applied to an odd integer $\sum \limits_{n > m}2^n + 2^m - 1$, an even integer $3\left(\sum \limits_{n > m}2^{n}\right) + 2^{m + 1} + 2^m - 2$ is obtained, which is exactly once divisible by 2, unless the lowest index reduces to zero. This implies that the sequence alternates between odd and even steps $m$ times. This governs the dynamics of the Collatz-type sequences because the value of $m$ determines the number of times the integer can be divided by 2 in each even step. A shortcut method is developed based on this dynamics that states that the even integer after $m$ odd-even steps are completed is $\left(\left(\frac{3}{2}\right)^m \sum \limits_{n > m}2^n\right) + 3^m - 1$. A shortcut method of this magnitude has never been utilized anywhere. The shortcut method for the modified Collatz sequence $5x + 1$ is also presented.
Keywords
Collatz; 3n+1
Subject
Computer Science and Mathematics, Algebra and Number Theory
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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