Solve the 3x+1 Problem

Jishe Feng

doi:10.20944/preprints202301.0163.v2

Preprint Article Version 2 Preserved in Portico This version is not peer-reviewed

Version 1 : Received: 6 January 2023 / Approved: 9 January 2023 / Online: 9 January 2023 (11:00:24 CET)
Version 2 : Received: 22 January 2023 / Approved: 23 January 2023 / Online: 23 January 2023 (09:33:04 CET)
Version 3 : Received: 27 January 2023 / Approved: 27 January 2023 / Online: 27 January 2023 (10:43:34 CET)
Version 4 : Received: 28 January 2023 / Approved: 30 January 2023 / Online: 30 January 2023 (09:23:25 CET)
Version 5 : Received: 14 February 2023 / Approved: 20 February 2023 / Online: 20 February 2023 (10:26:07 CET)

The 3x+1 problem asks the following: Suppose we start with a positive integer, and if it is odd then multiply it by 3 and add 1, and if it is even, divide it by 2. Then repeat this process as long as you can. Do you eventually reach the integer 1, no matter what you started with? Collatz conjecture (or 3n+1 problem) has been explored for about 85 years. In this article, we prove the Collatz conjecture by modifying Sharkovsky ordering of positive integers and denote the composition of the collatz function as a algebraic formula about $\frac{3^{m}}{(2^{r}}$, convert the problem to a algebraic problem, we can solve it completely.

The Collatz conjecture

Computer Science and Mathematics, Algebra and Number Theory

Comment 1

Received: 23 January 2023
Commenter: Jishe Feng
Commenter's Conflict of Interests: Author

Comment: We update Fig. 1 and Fig. 2, and give more detailed description of the Collatz graphs. correct many mistakes.

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Solve the 3x+1 Problem

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