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Finally a proof for the Collatz conjecture with the only possible five expressions of natural numbers which can respect the Collatz conjecture?


Problem/Ansatz:

I published a new short mathematical letter about the Collatz conjecture which is related to my previous article in this topic. This short article gives the only possible five expressions of natural numbers which can respect the Collatz conjecture. Can you now find the counterexample by using the results or prove that the Collatz conjecture is true ?
Here is the link:
https://www.researchgate.net/publication/377746599_The_only_five_expressions_of_numbers_which_respect_the_Collatz_conjecture

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