A Solution of The Collatz Conjecture Problem

Baoyuan Duan

doi:10.20944/preprints202301.0541.v14

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preprints.org > computer science and mathematics > signal processing > doi: 10.20944/preprints202301.0541.v14

Preprint Article Version 14 Preserved in Portico This version is not peer-reviewed

A Solution of The Collatz Conjecture Problem

Baoyuan Duan

Version 1 : Received: 30 January 2023 / Approved: 30 January 2023 / Online: 30 January 2023 (06:21:02 CET)
Version 2 : Received: 11 February 2023 / Approved: 13 February 2023 / Online: 13 February 2023 (02:53:06 CET)
Version 3 : Received: 4 March 2023 / Approved: 6 March 2023 / Online: 6 March 2023 (04:14:16 CET)
Version 4 : Received: 11 March 2023 / Approved: 13 March 2023 / Online: 13 March 2023 (03:05:38 CET)
Version 5 : Received: 28 March 2023 / Approved: 28 March 2023 / Online: 28 March 2023 (05:32:08 CEST)
Version 6 : Received: 3 April 2023 / Approved: 3 April 2023 / Online: 3 April 2023 (07:22:34 CEST)
Version 7 : Received: 10 April 2023 / Approved: 11 April 2023 / Online: 11 April 2023 (03:28:27 CEST)
Version 8 : Received: 22 June 2023 / Approved: 25 June 2023 / Online: 25 June 2023 (04:40:44 CEST)
Version 9 : Received: 20 July 2023 / Approved: 21 July 2023 / Online: 21 July 2023 (08:53:32 CEST)
Version 10 : Received: 10 August 2023 / Approved: 10 August 2023 / Online: 11 August 2023 (03:01:11 CEST)
Version 11 : Received: 19 September 2023 / Approved: 20 September 2023 / Online: 21 September 2023 (03:25:23 CEST)
Version 12 : Received: 14 October 2023 / Approved: 17 October 2023 / Online: 17 October 2023 (07:08:50 CEST)
Version 13 : Received: 28 October 2023 / Approved: 30 October 2023 / Online: 30 October 2023 (09:47:16 CET)
Version 14 : Received: 19 November 2023 / Approved: 21 November 2023 / Online: 21 November 2023 (10:43:13 CET)
Version 15 : Received: 9 April 2024 / Approved: 9 April 2024 / Online: 10 April 2024 (09:37:50 CEST)
Version 16 : Received: 20 April 2024 / Approved: 22 April 2024 / Online: 23 April 2024 (09:43:39 CEST)

How to cite: Duan, B. A Solution of The Collatz Conjecture Problem. Preprints 2023, 2023010541. https://doi.org/10.20944/preprints202301.0541.v14 Duan, B. A Solution of The Collatz Conjecture Problem. Preprints 2023, 2023010541. https://doi.org/10.20944/preprints202301.0541.v14

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MDPI and ACS Style

Duan, B. A Solution of The Collatz Conjecture Problem. Preprints 2023, 2023010541. https://doi.org/10.20944/preprints202301.0541.v14

APA Style

Duan, B. (2023). A Solution of The Collatz Conjecture Problem. Preprints. https://doi.org/10.20944/preprints202301.0541.v14

Chicago/Turabian Style

Duan, B. 2023 "A Solution of The Collatz Conjecture Problem" Preprints. https://doi.org/10.20944/preprints202301.0541.v14

Abstract

Build a special identical equation, use its calculation characters to prove and search for solution of any odd converging to 1 equation through (*3+1)/2^k operation, change the operation to (*3+2^m-1)/2^k, get a solution for this equation. Furthermore, analysis the sequences produced by iteration calculation during the procedure of searching for solution, build a weight function model, prove it monotonically decreases, build a complement weight function model, prove it has many chances to increase to its convergence state. Build a (*3+2^m-1)/2^k odd tree, prove if odd in (*3+2^m-1)/2^k long huge odd sequence can not converge, the sequence must walk out of the boundary of the tree after infinite steps of (*3+2^m-1)/2^k operation.

Keywords

Collatz conjecture; (*3+1)/2^k odd sequence; (*3+2^m-1)/2^k odd sequence; (*3+2^m-1)/2^k odd tree; weight function

Subject

Computer Science and Mathematics, Signal Processing

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 (4)

Comment 1

Received: 21 November 2023
Commenter: baoyuan duan
Commenter's Conflict of Interests: Author

Comment: Enhance the proof.

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Comment 2

Received: 23 January 2024
Commenter:

baoyuan duan

Commenter's Conflict of Interests: I am one of the author

Comment: Pen error in page 3. 'form (k+j=odd)' should be 'form (k+j=even)'

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Comment 3

Received: 9 March 2024
Commenter:

baoyuan duan

Commenter's Conflict of Interests: I am one of the author

Comment: It is better to insert $s_{(1,2)}-s_{(2)}>0$ and $s_{(1,1,2)}-s_{(2)}>0$ in section 6.2.

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Comment 4

Received: 22 March 2024
Commenter:

baoyuan duan

Commenter's Conflict of Interests: I am one of the author

Comment: How to expand loop case?Suppose exists $b_1,b_2...b_i,b_1,b_2...b_i$ (*3+1)/2^k odd sequence, $a_1,a_2...a_i,a_{i+1},a_{i+2}...a_{2i}$ is corresponding (*3+2^m-1)/2^k odd sequence.$a_{i+1}$ is not possible equal to $a_1$ because any $a_j$ adds one 1 to binary head within 3 (*3+2^m-1)/2^k steps. The loop sequence must include 2 or more $(2^+)$ steps because any binary Collatz odd '11...1'(or part of odd is this form) can not change back to itself by only one $(2^+)$ step.These are not introduced in the article, but are obvious.

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