Preprint Article Version 5 Preserved in Portico This version is not peer-reviewed

# Solve the 3x+1 Problem by the Multiplication and Division of Binary Numbers

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)

How to cite: Feng, J. Solve the 3x+1 Problem by the Multiplication and Division of Binary Numbers. Preprints 2023, 2023010163. https://doi.org/10.20944/preprints202301.0163.v5 Feng, J. Solve the 3x+1 Problem by the Multiplication and Division of Binary Numbers. Preprints 2023, 2023010163. https://doi.org/10.20944/preprints202301.0163.v5

## Abstract

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 paper, we convert an integer number from decimal to binary number, and convert the Collatz function to binary function, which is multiplication and division of two binary numbers. Finally the iternation of the Collatz function, eventually reach the integer 1, thus we solve the 3x+1 problem completely.

## Keywords

The Collatz conjecture

## Subject

Computer Science and Mathematics, Algebra and Number Theory

Comment 1
Commenter: Jishe Feng
Commenter's Conflict of Interests: Author
Comment: We rewrite and recorrect.
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