The Collatz Conjecture, also known as the $3X + 1$ conjecture, involves taking any positive integer and following a specific set of steps to generate a sequence. The conjecture claims that no matter which positive integer you start with, you will always eventually reach the number $1$. The steps ...

Setup: Let an odd integer $\mathcal{X}$ be represented in the modified form as $$\{\sum \limits_{M>m} b_M 2^M\} + 2^m - 1$$ where $b_M \in {0,1}$. I have an insight that all the repeating odd integers in the Collatz-type 5Z+1 sequence end in either $2^1 - 1$ or $2^2 - 1$. For example: 13 is $2^3+...