Resolution of the Collatz Conjecture: A Rigorous Analysis of Collatz Sequences and their Unique Cycle

Eduardo Diedrich

doi:10.20944/preprints202406.0256.v4

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preprints.org > computer science and mathematics > mathematics > doi: 10.20944/preprints202406.0256.v4 (registering DOI)

Preprint Article Version 4 This version is not peer-reviewed

Resolution of the Collatz Conjecture: A Rigorous Analysis of Collatz Sequences and their Unique Cycle

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)

How to cite: Diedrich, E. Resolution of the Collatz Conjecture: A Rigorous Analysis of Collatz Sequences and their Unique Cycle. Preprints 2024, 2024060256. https://doi.org/10.20944/preprints202406.0256.v4 Diedrich, E. Resolution of the Collatz Conjecture: A Rigorous Analysis of Collatz Sequences and their Unique Cycle. Preprints 2024, 2024060256. https://doi.org/10.20944/preprints202406.0256.v4

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MDPI and ACS Style

Diedrich, E. Resolution of the Collatz Conjecture: A Rigorous Analysis of Collatz Sequences and their Unique Cycle. Preprints 2024, 2024060256. https://doi.org/10.20944/preprints202406.0256.v4

APA Style

Diedrich, E. (2024). Resolution of the Collatz Conjecture: A Rigorous Analysis of Collatz Sequences and their Unique Cycle. Preprints. https://doi.org/10.20944/preprints202406.0256.v4

Chicago/Turabian Style

Diedrich, E. 2024 "Resolution of the Collatz Conjecture: A Rigorous Analysis of Collatz Sequences and their Unique Cycle" Preprints. https://doi.org/10.20944/preprints202406.0256.v4

Abstract

This article presents a rigorous approach to the Collatz Conjecture, focusing on fundamental properties of Collatz sequences. We establish key properties of the Collatz function and its inverse, including surjectivity and injectivity. The structure of Collatz sequences is analyzed in depth, proving important results such as the Bounded Subsequence Property and the uniqueness of cycles. Central theorems on the properties of Collatz sequences, including the boundedness of all sequences and the nature of the unique cycle, are presented and proved. These results culminate in a complete resolution of the Collatz Conjecture, demonstrating that all Collatz sequences eventually reach the cycle {1, 4, 2}. We provide a rigorous proof of the conjecture, while emphasizing the need for thorough peer review and verification by the mathematical community given the significance of this long-standing problem.

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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