Preprint Article Version 11 Preserved in Portico This version is not peer-reviewed

A Solution of The Collatz Conjecture Problem

baoyuan duan
* ORCID logo
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.v11 duan, B. A Solution of The Collatz Conjecture Problem. Preprints 2023, 2023010541. https://doi.org/10.20944/preprints202301.0541.v11

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 decrease progressively to 0, build a complement weight function model, prove it 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.

Download PDF

Comments (6)

Comment 1
Received: 21 September 2023
Commenter: baoyuan duan
Commenter's Conflict of Interests: Author
Comment: make the proof clearer
+ Respond to this comment
Comment 2
Received: 22 September 2023
Commenter:
baoyuan duan

Commenter's Conflict of Interests: I am one of the author
Comment: Modify: Now rebuild, do not change the first step and the last step, we move number property 2^k of all other downward steps next to the first step(suppose we can move in order to compare, we can do it from math calculation angle), then forward steps and upward steps....Then we merge the last step, upward steps before it, forward steps and the last downward step before forward steps to one step. The new sequence(has l-1, l or l+1 steps) must have a “transform position” increment ratio>3/4...
The change does not affect the conclusion.
+ Respond to this comment
Response 1 to Comment 2
Received: 26 September 2023
Commenter:
baoyuan duan

Commenter's Conflict of Interests: I am one of the author
Comment: Sorry, it is not necessary to modify.
Comment 3
Received: 30 September 2023
Commenter:
baoyuan duan

Commenter's Conflict of Interests: I am one of the author
Comment: pen error in page 12: "the transform positions of equivalent elements(add binary 1s in head) of elements in left half part in m-1 layer are all in right half part in m-1 layer" should be "the equivalent elements of elements in left half part in m-1 layer(by adding binary 1s in head) are all in right half part in m layer"
+ Respond to this comment
Response 1 to Comment 3
Received: 1 October 2023
Commenter:
baoyuan duan

Commenter's Conflict of Interests: I am one of the author
Comment: should be "the equivalent elements of elements in m-1 layer(by adding binary 1s in head) are all in right half part in m layer"
Comment 4
Received: 3 October 2023
Commenter:
baoyuan duan

Commenter's Conflict of Interests: I am one of the author
Comment: pen errors in page 12: "cases of (upward, upward), (downward, downward), (upward, downward), (upward, forward) ratio >3/4;" should be "cases of (upward, upward), (downward, downward), (upward, downward), (upward, forward), (forward, downward) ratio >3/4;". "the transformed positions of equivalence elements(add binary 1s in head) of elements in left half part in m-1 layer are all in right half part in m-1 layer" should be "the equivalent elements of elements in m-1 layer(by adding binary 1s in head) are all in right half part in m layer"
+ Respond to this comment

We encourage comments and feedback from a broad range of readers. See criteria for comments and our Diversity statement.

Leave a public comment
Send a private comment to the author(s)
* All users must log in before leaving a comment
Views 0
Downloads 0
Comments 6
Metrics 0
Get PDF Cite Share 0
Bookmark BibSonomy Mendeley Reddit Delicious
×
Alerts
Notify me about updates to this article or when a peer-reviewed version is published.
We use cookies on our website to ensure you get the best experience.
Read more about our cookies here.