The Collatz Conjecture: A New Proof using Algebraic Inverse Trees

Eduardo Diedrich

doi:10.20944/preprints202310.0773.v13

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preprints.org > computer science and mathematics > mathematics > doi: 10.20944/preprints202310.0773.v13

Preprint Article Version 13 Preserved in Portico This version is not peer-reviewed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)

A peer-reviewed article of this Preprint also exists.

Diedrich, E. The Collatz Conjecture: A New Proof Using Algebraic Inverse Tree. International Journal of Pure and Applied Mathematics Research 2024, 4, 34–79, doi:10.51483/ijpamr.4.1.2024.34-79. Diedrich, E. The Collatz Conjecture: A New Proof Using Algebraic Inverse Tree. International Journal of Pure and Applied Mathematics Research 2024, 4, 34–79, doi:10.51483/ijpamr.4.1.2024.34-79.

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: 29 December 2023
Commenter: Eduardo Diedrich
Commenter's Conflict of Interests: Author

Comment: The current version has significantly increased in size, expanding from 22 to 98 pages.
Numerous new sections have been added, highlighting:
Analysis of the need for new approaches to the Conjecture (Section 3.2)
Introductory topological concepts (Section 4)
Formalization of the Collatz function and its inverse (Section 6)
Definitions and fundamental properties of Algebraic Inverse Trees (Section 7)
Extension to the general case of recursive functions(Section 8)
Expansion on potential practical applicationsAdditional details about the proof

The existing sections have also been significantly expanded. For example:
More historical context and relevance (Section 3.1)
A more comprehensive comparison with other approaches (Section 3)
More complete formalizations on topology (Section 5)
A deeper analysis of special cases (Section 7)
Overall, there is a much higher level of mathematical formalization noted, with an extensive introduction of formal definitions, lemmas, theorems, and their associated proofs. New sections on empirical and experimental validation are also included.In summary, the updated version presents significant expansion and deepening in both scope and the level of mathematical formalism, resulting in a much more comprehensive and detailed exposition on the use of Algebraic Inverse Trees to address the Collatz Conjecture.

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