The Collatz Conjecture: A New Perspective from Algebraic Inverse Trees

Eduardo Diedrich

doi:10.20944/preprints202310.0773.v2

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preprints.org > computer science and mathematics > mathematics > doi: 10.20944/preprints202310.0773.v2

Preprint Article Version 2 Preserved in Portico This version is not peer-reviewed

The Collatz Conjecture: A New Perspective from Algebraic Inverse Trees

Eduardo Diedrich

Version 1 : Received: 11 October 2023 / Approved: 12 October 2023 / Online: 12 October 2023 (04:43:40 CEST)
Version 2 : Received: 13 October 2023 / Approved: 13 October 2023 / Online: 13 October 2023 (04:48:03 CEST)
Version 3 : Received: 13 October 2023 / Approved: 17 October 2023 / Online: 17 October 2023 (10:26:40 CEST)
Version 4 : Received: 17 October 2023 / Approved: 18 October 2023 / Online: 18 October 2023 (10:09:49 CEST)
Version 5 : Received: 19 October 2023 / Approved: 20 October 2023 / Online: 20 October 2023 (10:28:38 CEST)
Version 6 : Received: 21 October 2023 / Approved: 23 October 2023 / Online: 23 October 2023 (10:35:42 CEST)
Version 7 : Received: 26 October 2023 / Approved: 26 October 2023 / Online: 27 October 2023 (09:14:36 CEST)
Version 8 : Received: 28 October 2023 / Approved: 30 October 2023 / Online: 30 October 2023 (09:38:23 CET)
Version 9 : Received: 30 October 2023 / Approved: 31 October 2023 / Online: 31 October 2023 (11:01:44 CET)
Version 10 : Received: 1 November 2023 / Approved: 2 November 2023 / Online: 2 November 2023 (08:16:04 CET)
Version 11 : Received: 8 November 2023 / Approved: 8 November 2023 / Online: 8 November 2023 (07:53:02 CET)
Version 12 : Received: 21 November 2023 / Approved: 22 November 2023 / Online: 24 November 2023 (04:11:49 CET)
Version 13 : Received: 27 December 2023 / Approved: 28 December 2023 / Online: 29 December 2023 (01:14:06 CET)
Version 14 : Received: 16 March 2024 / Approved: 17 March 2024 / Online: 18 March 2024 (10:42:38 CET)
Version 15 : Received: 24 March 2024 / Approved: 25 March 2024 / Online: 26 March 2024 (11:50:16 CET)
Version 16 : Received: 28 March 2024 / Approved: 28 March 2024 / Online: 29 March 2024 (11:23:27 CET)
Version 17 : Received: 30 March 2024 / Approved: 1 April 2024 / Online: 2 April 2024 (11:53:22 CEST)
Version 18 : Received: 28 April 2024 / Approved: 29 April 2024 / Online: 29 April 2024 (09:46:13 CEST)

How to cite: Diedrich, E. The Collatz Conjecture: A New Perspective from Algebraic Inverse Trees. Preprints 2023, 2023100773. https://doi.org/10.20944/preprints202310.0773.v2 Diedrich, E. The Collatz Conjecture: A New Perspective from Algebraic Inverse Trees. Preprints 2023, 2023100773. https://doi.org/10.20944/preprints202310.0773.v2

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MDPI and ACS Style

Diedrich, E. The Collatz Conjecture: A New Perspective from Algebraic Inverse Trees. Preprints 2023, 2023100773. https://doi.org/10.20944/preprints202310.0773.v2

APA Style

Diedrich, E. (2023). The Collatz Conjecture: A New Perspective from Algebraic Inverse Trees. Preprints. https://doi.org/10.20944/preprints202310.0773.v2

Chicago/Turabian Style

Diedrich, E. 2023 "The Collatz Conjecture: A New Perspective from Algebraic Inverse Trees" Preprints. https://doi.org/10.20944/preprints202310.0773.v2

Abstract

This paper addresses the Collatz Conjecture, an open question in mathematics that postulates all positive integers will eventually reach one when a pair of specific operations are repeatedly applied. Despite its apparent simplicity, the conjecture lacks a formal proof. To tackle this enigma, we introduce Algebraic Inverse Trees (AITs), data structures that trace inverse operations of the Collatz sequence. This new approach not only elaborates our unique methodology but also sheds light on the underlying complexities of the Collatz Conjecture.

Keywords

Collatz Cojecture; Algebraic Inverse Trees; Hypotesis of Representations; Hypotesis of Saturation; Suryectivity of Reverse Function

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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Comments (1)

Comment 1

Received: 13 October 2023
Commenter: Eduardo Diedrich
Commenter's Conflict of Interests: Author

Comment: Header Restructuring: I modified the header to "Algebraic Inverse Trees (AITs) for Analyzing the Collatz Sequence" to provide a clearer idea of the subsection's focus.
Consolidation of Introduction: Instead of several standalone sentences about AITs, I grouped the information into a more cohesive introductory paragraph that gives an overarching understanding of AITs and their purpose.
Use of itemize for lists: I replaced sections that used the \textbf{} format for lists with LaTeX's itemize environment. This provides a clearer presentation and eases readability.
Simplified Description: I simplified and clarified the description about the "even" and "odd" parents in the "Multiple Parents in AITs" section.
Redundancy Reduction: I eliminated repetitions and presented the information in a more direct and concise manner, especially in the "Constructing AITs" section.
Reordering: I shuffled some points around to enhance logical flow and cohesion in the content.
Refinement of Mathematical Notation: I ensured mathematical notation and formulas were consistent and clear throughout.

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