Number Theory

    

The Non-Existence of Non-Trivial Periodic Orbitsin the Collatz Mapping

Authors: Xiaofeng Hu

The Collatz conjecture states that for any given positive integer N,if N is even, divide it by 2; if N is odd, multiply it by 3 and add 1.Repeating this process,N will eventually become 1. This paper provesthat any positive odd integer O other than 1 cannot return to itself nomatter how many times the iteration is performed. We derive the generalformula satisfying this condition and rigorously prove by mathematicalinduction that this formula equals 1 uniquely in the set of positive oddintegers. We thus conclude that there are no non-trivial periodic orbitsin the Collatz mapping.

Comments: 6 Pages. (Note by viXra Admin: Please cite and list scientific reference and submit article written with AI assistance to ai.viXra.org)

Download: PDF

Submission history

[v1] 2026-04-17 00:26:11

Unique-IP document downloads: 99 times

Vixra.org is a pre-print repository rather than a journal. Articles hosted may not yet have been verified by peer-review and should be treated as preliminary. In particular, anything that appears to include financial or legal advice or proposed medical treatments should be treated with due caution. Vixra.org will not be responsible for any consequences of actions that result from any form of use of any documents on this website.

Add your own feedback and questions here:
You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.

comments powered by Disqus