Abstract
The objective of this article is to demonstrate the Collatz Conjecture through the Sets and Binary Numbers Theory, in this manner: 2<sup>n</sup> + 2<sup>n-1</sup>+...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 2<sup>n</sup>. 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 2<sup>n</sup> 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.
Highlights
In this article, the Theory of Sets and Binary Numbers will be used, with the ED [Portuguese acronym] (Written by Definition) method, to investigate the Collatz conjecture through the results obtained
This paper demonstrates how to obtain numbers in which the 3n+1 operation does not take place at all, takes place only once, or in which this operation appears at least "n" times
Lothar Collatz conjectured that, by pursuing these operations, one will always arrive at the number 1
Summary
The Theory of Sets and Binary Numbers will be used, with the ED [Portuguese acronym] (Written by Definition) method, to investigate the Collatz conjecture through the results obtained. The Collatz Conjecture, or 3n+1 problem, was formulated in 1937 by German mathematician, Lothar Collatz. It is a mathematical assumption which is thought to be true, but has yet to be proven or rejected. The Collatz Conjecture asserts that, by performing the following operations: begin with a natural number. If this number is even, divide by 2. Lothar Collatz conjectured that, by pursuing these operations, one will always arrive at the number 1
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