Number Theory

    

A Proof of the Collatz (3x+1) Conjecture

Authors: Xingyuan Zhang

In this paper we had given an elementary proof of the Collatz conjecture, it holds. By detailed analysis of the properties of both forward and inverse operations of the proposition, we had some important conclusions: 1, there are no cycles except 1 to 1, and for a given odd it either goes to infinity or returns to 1 in forward operations; 2, there hasn’t any triple in the forward path numbers; 3, there have an infinity number of inverse path numbers which had been defined as similar numbers between them in one time of inverse operation; 4, to do inverse operations (defined as reverse tracing) repeatedly from the odd 1, it will obtain all of the odds; 5, for any odd obtained by tracing, to do forward operations, it must return to 1 along the reverse tracing path.

Comments: 21 Pages. This article did proved Collatz conjecture. Method is reverse tracing. My conclusions and middle results are very beautiful. If interested, to discuss, or even cooperate to finish the paper for submission!

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Submission history

[v1] 2023-01-29 06:19:56
[v2] 2023-03-02 08:19:43
[v3] 2023-03-17 12:29:45
[v4] 2023-03-23 06:25:32
[v5] 2023-08-30 08:53:46

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