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The function C is defined on positive integers as follows:

\(C(n) = \begin{cases} \dfrac n 2 & \text{if $n$ is even}, \\ 3n+1 & \text{if $n$ is odd}. \end{cases}\)
Find the smallest positive integer of m such that C^m(9)=1.

 Apr 30, 2023
 #1
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The smallest m that works is 5.

 May 1, 2023
 #2
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C^m(x)=C(C(C(C ... (x))))..., performing C m times. Do this to 9:

9 is odd, C(9)=28.

28 is odd, so C(C(9))=14.

14 is even, so C(C(C(9)))=7. 

Continue to get 22,11,34,17,52,26,13,40,20,10,5,16,8,4,2,1

After applying C 19 times, we have reached 1. Therefore, the smallest m is 19.

 May 2, 2023

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