Version 1
: Received: 20 May 2021 / Approved: 21 May 2021 / Online: 21 May 2021 (07:52:14 CEST)
Version 2
: Received: 1 November 2021 / Approved: 3 November 2021 / Online: 3 November 2021 (19:57:42 CET)
Version 3
: Received: 23 February 2022 / Approved: 24 February 2022 / Online: 24 February 2022 (08:38:01 CET)

How to cite:
Jiang, L. 3x + 1 Problem: Iterative Operations Neither Cause Loops nor Diverge to Infinity. Preprints2021, 2021050499 (doi: 10.20944/preprints202105.0499.v2).
Jiang, L. 3x + 1 Problem: Iterative Operations Neither Cause Loops nor Diverge to Infinity. Preprints 2021, 2021050499 (doi: 10.20944/preprints202105.0499.v2).

Cite as:

Jiang, L. 3x + 1 Problem: Iterative Operations Neither Cause Loops nor Diverge to Infinity. Preprints2021, 2021050499 (doi: 10.20944/preprints202105.0499.v2).
Jiang, L. 3x + 1 Problem: Iterative Operations Neither Cause Loops nor Diverge to Infinity. Preprints 2021, 2021050499 (doi: 10.20944/preprints202105.0499.v2).

Abstract

The 3x+1 problem is a problem of continuous iteration for integers. According to the basic theorem of arithmetic and the way of iteration, we derive a general formula for continuous iteration for odd integers. Through this formula, we can construct a loop iteration equation and obtain the result of the equation: the equation has only one positive integer solution. In addition, this general formula can be converted into a linear indeterminate equation. The process of solving this equation shows that the relationship between the iteration result and the odd number being iterated is linear. Extending this result to all positive even numbers, we get the answer to the 3x + 1 question.

MATHEMATICS & COMPUTER SCIENCE, Algebra & Number Theory

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.

Received:
3 November 2021
Commenter:
Jiang Li
Commenter's Conflict of Interests:
Author

Comment:
The 3x+1 problem is a problem of continuous iteration for integers. Ac- cording to the original description of the problem, we derive a formula that can perform continuous iterative operations on odd numbers. We can convert this formula into a linear equation. The process of solving this equation shows that the relationship between the it- eration result and the odd number being iterated is linear. In addition, we can construct a loop iteration equation by the formula and obtain the result of the equation: the equation has only one positive integer solution. Extending this result to all positive even numbers, we get the answer to the 3x + 1 question

Commenter: Jiang Li

Commenter's Conflict of Interests: Author

cording to the original description of the problem, we derive a formula that can perform

continuous iterative operations on odd numbers. We can convert this formula into a linear

equation. The process of solving this equation shows that the relationship between the it-

eration result and the odd number being iterated is linear. In addition, we can construct a

loop iteration equation by the formula and obtain the result of the equation: the equation

has only one positive integer solution. Extending this result to all positive even numbers,

we get the answer to the 3x + 1 question