The Collatz Conjecture: A Resolution through Inverse Function Generative Completeness

Eduardo Diedrich

doi:10.20944/preprints202406.0256.v7

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preprints.org > computer science and mathematics > mathematics > doi: 10.20944/preprints202406.0256.v7

Preprint Article Version 7 Preserved in Portico This version is not peer-reviewed

The Collatz Conjecture: A Resolution through Inverse Function Generative Completeness

Eduardo Diedrich

Version 1 : Received: 1 June 2024 / Approved: 4 June 2024 / Online: 5 June 2024 (10:19:24 CEST)
Version 2 : Received: 21 June 2024 / Approved: 21 June 2024 / Online: 24 June 2024 (08:08:06 CEST)
Version 3 : Received: 12 July 2024 / Approved: 15 July 2024 / Online: 15 July 2024 (09:50:14 CEST)
Version 4 : Received: 2 August 2024 / Approved: 5 August 2024 / Online: 6 August 2024 (02:32:33 CEST)
Version 5 : Received: 7 August 2024 / Approved: 7 August 2024 / Online: 8 August 2024 (03:57:55 CEST)
Version 6 : Received: 9 August 2024 / Approved: 12 August 2024 / Online: 13 August 2024 (03:11:01 CEST)
Version 7 : Received: 19 August 2024 / Approved: 19 August 2024 / Online: 19 August 2024 (10:52:12 CEST)

How to cite: Diedrich, E. The Collatz Conjecture: A Resolution through Inverse Function Generative Completeness. Preprints 2024, 2024060256. https://doi.org/10.20944/preprints202406.0256.v7 Diedrich, E. The Collatz Conjecture: A Resolution through Inverse Function Generative Completeness. Preprints 2024, 2024060256. https://doi.org/10.20944/preprints202406.0256.v7

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MDPI and ACS Style

Diedrich, E. The Collatz Conjecture: A Resolution through Inverse Function Generative Completeness. Preprints 2024, 2024060256. https://doi.org/10.20944/preprints202406.0256.v7

APA Style

Diedrich, E. (2024). The Collatz Conjecture: A Resolution through Inverse Function Generative Completeness. Preprints. https://doi.org/10.20944/preprints202406.0256.v7

Chicago/Turabian Style

Diedrich, E. 2024 "The Collatz Conjecture: A Resolution through Inverse Function Generative Completeness" Preprints. https://doi.org/10.20944/preprints202406.0256.v7

Abstract

This article presents a novel resolution of the Collatz Conjecture, centered on the concept of Generative Completeness of the inverse Collatz function. We introduce and rigorously prove that for all N ∈ ℕ⁺, there exists a minimal generator m_N = 1 such that all positive integers up to N can be generated through successive applications of the inverse Collatz function G. This key property, which we term "Generative Completeness", forms the cornerstone of our proof. Building upon this foundation, we establish several crucial results: The boundedness of all Collatz sequences The existence and uniqueness of cycles in Collatz sequences The nature of the unique cycle as {1, 4, 2} We then present three distinct approaches to resolving the Collatz Conjecture, all fundamentally rooted in the Generative Completeness property. These diverse methods not only prove the conjecture but also provide deep insights into the structure of Collatz sequences.

Keywords

Collatz conjecture; 3x+1 problem; number theory; sequence analysis; cycle properties; inverse Collatz function; boundedness; divergence; mathematical induction; proof techniques

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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