Some extensions of Collatz (periodic) conjecture

The Collatz conjecture is a famous conjecture aiming to estimate the behaviors of positive integers after a series of iterations of the Collatz function f(n), where f(n) = {3n+1/2 if n is odd, n/2 if n is even. Collatz conjecture has attracted a great deal of attention since it was put forward in 1930s due to the surprising nature. However, after strenuous efforts, many mathematicians think the current knowledge seems insufficient to solve Collatz conjecture, although it appears in a very simple form. As a replacement, a "periodic" (weaker) version of Collatz conjecture (called Collatz periodic conjecture) was proposed and discussed. In this paper, we would like to explore what properties Collatz (periodic) conjecture could imply. Maybe it is another possible direction to confirm or disprove Collatz (periodic) conjecture.