A Grouping Method That Precisely Explains Value Changes And Prime Factor Changes Of Collatz Sequences And Proves That Almost All Collatz Sequences Attain 1

Abstract

This article proposes a method of grouping odd numbers into a closed group for Collatz Transformations, along with the finding that each odd number in the group is associated with an even number which determines the value changes of that odd number. By using this method we can *precisely* explain and partially predict the value changes and the prime factor changes of all Collatz sequences. The article also gives out the probabilities of the distribution of prime factor 2s which is essential in explaining and predicting the changes of Collatz sequences, and the probabilities of the increasing and decreasing speed of Collatz sequences, along with the proof of the conclusion that almost all Collatz sequences will attain 1.