Proving the Collatz Conjecture with Binaries Numbers

Abstract

The objective of this article is to demonstrate the Collatz Conjecture through the Sets and Binary Numbers Theory, in this manner: 2n + 2n-1+...1. This study shows that there are subsequences of odd numbers within the Collatz sequences, and that by proving the proposition is true for these subsequences, it is subsequently proven that the entire proposition is correct. It is also proven that a sequence which begins with a natural number is generated by a set of operations: Multiplication by 3, addition of 1 and division by 2n. This set of operations shall be called “Movement” in this study, and may be increasing when n=1, and decreasing for n ≥ 2. The numbers in 2n form generate decreasing sequences in which the 3n+1 operation does not occur. One of the important discoveries is how to generate numbers in which the 3n+1 operation only occurs once and how to generate numbers with a minimum quantity of increasing movements that are the numbers of greater “orbits” (Longer sequences that take longer to reach the number one). The conclusion is that, as the decreasing numbers dominate as compared to the increasing ones, the statement that the sequence is always going to reach the number 1 is true.