Preprint Article Version 2 Preserved in Portico This version is not peer-reviewed

# 3x + 1 Problem: Iterative Operations Neither Cause Loops nor Diverge to Infinity

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. Preprints 2021, 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.

## Keywords

3x + 1 problem; Collatz conjecture; Syracuse problem; iteration

## Subject

MATHEMATICS & COMPUTER SCIENCE, Algebra & Number Theory

Comment 1
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
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