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Proving the Collatz Conjecture with Binaries Numbers
Pure and Applied Mathematics Journal
Volume 7, Issue 5, October 2018, Pages: 68-77
Received: Oct. 13, 2018; Accepted: Nov. 22, 2018; Published: Dec. 24, 2018
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Olinto de Oliveira Santos, CETEP- Territorial Center for Education Professional of the Coast of the Discovery, Eunápolis, Brazil
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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.
Binary Numbers, Collatz Conjecture, Hail Sequences
To cite this article
Olinto de Oliveira Santos, Proving the Collatz Conjecture with Binaries Numbers, Pure and Applied Mathematics Journal. Vol. 7, No. 5, 2018, pp. 68-77. doi: 10.11648/j.pamj.20180705.12
Copyright © 2018 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License ( which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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