The Collatz Conjecture: A New Proof using Algebraic Inverse Trees

Eduardo Diedrich

doi:10.20944/preprints202310.0773.v12

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preprints.org > computer science and mathematics > mathematics > doi: 10.20944/preprints202310.0773.v12

Preprint Article Version 12 Preserved in Portico This version is not peer-reviewed

The Collatz Conjecture: A New Proof using 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 Proof using Algebraic Inverse Trees. Preprints 2023, 2023100773. https://doi.org/10.20944/preprints202310.0773.v12 Diedrich, E. The Collatz Conjecture: A New Proof using Algebraic Inverse Trees. Preprints 2023, 2023100773. https://doi.org/10.20944/preprints202310.0773.v12

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MDPI and ACS Style

Diedrich, E. The Collatz Conjecture: A New Proof using Algebraic Inverse Trees. Preprints 2023, 2023100773. https://doi.org/10.20944/preprints202310.0773.v12

APA Style

Diedrich, E. (2023). The Collatz Conjecture: A New Proof using Algebraic Inverse Trees. Preprints. https://doi.org/10.20944/preprints202310.0773.v12

Chicago/Turabian Style

Diedrich, E. 2023 "The Collatz Conjecture: A New Proof using Algebraic Inverse Trees" Preprints. https://doi.org/10.20944/preprints202310.0773.v12

Abstract

This paper presents a novel approach to demonstrate the Collatz Conjecture, an unsolved problem in mathematics stating that all positive integers will eventually converge to 1 when subjected to a recursive sequence of operations. We introduce Algebraic Inverse Trees (AITs), innovative data structures that characterize relationships within the Collatz sequence by tracing inverse transformations back to the origin. Leveraging properties unveiled through this inverted representation, including absence of non-trivial cycles and guaranteed convergence of paths, we construct a topological framework to formally prove that all trajectories invariably lead to 1 under the Collatz iterative process. This modern technique provides not only the machinery to manifest a rigorous proof, but also grants intuitive insights into the deep complexity of this notorious yet elementary conjecture.

Keywords

Collatz Conjecture; Algebraic Inverse Trees; Formal Proof; Topological Framework; Inverse Graph Characterization; Path Convergence; Cycles Absence; Recursive Number Sequences; Discrete Mathematics Structures

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: 24 November 2023
Commenter: Eduardo Diedrich
Commenter's Conflict of Interests: Author

Comment: Efforts have been made to create the deductive logical system for the proof of the Collatz Conjecture as complete and simple as possible, and the theorems have been organized thematically. The structural and topological equivalence between Collatz sequences and AIT has been successfully demonstrated

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