Number Theory

    

The Graphical Method of the Kakutani’s Problem

Authors: Xingyuan Zhang

In this paper we had given another elementary proof of the Kakutani’s problem by using the Kakutani’s Angle, it holds. By detailed analysis of the properties of both forward and inverse operations of the proposition, we had some important conclusions: 1, there hasn’t any triple in the forward path numbers; 2, there have an infinity number of inverse path numbers which had been defined as similar numbers in one time of inverse operation; 3, on the figure of Kakutani’s Angle, the operation path of any odd is unique; 4, the inverse operations can start with any odd, and all of the path numbers on the countless paths is getting larger and larger, on the contrary, to do forward operations for any inverse path number, it must go back to the starting point or to 1. It’s not difficult to prove if understanding its operational mechanism and grasping the methods.

Comments: 10 pages. This is my second proof of the Collatz conjecture called graphical method. It can be only 6 pages. It has also other solutions or ideas, such as only using a table and in binary. My first proof is at viXra: 2301.0154.

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Submission history

[v1] 2023-05-18 08:35:19

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