Proving the Collatz Conjecture with Binaries Numbers

Olinto de Oliveira Santos

doi:doi:10.11648/j.pamj.20180705.12

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Proving the Collatz Conjecture with Binaries Numbers

Olinto de Oliveira Santos

Published in Pure and Applied Mathematics Journal (Volume 7, Issue 5)

Received: 13 October 2018 Accepted: 22 November 2018 Published: 24 December 2018

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Abstract

The objective of this article is to demonstrate the Collatz Conjecture through the Sets and Binary Numbers Theory, in this manner: 2ⁿ + 2^n-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.

Published in	Pure and Applied Mathematics Journal (Volume 7, Issue 5)
DOI	10.11648/j.pamj.20180705.12
Page(s)	68-77
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2018. Published by Science Publishing Group

Keywords

Binary Numbers, Collatz Conjecture, Hail Sequences

References

[1]	FILHO, A. C. S. (2010) “Formulações Alternativas para Conjectura de Collatz”. XI Encontro de Pesquisadores. pp. 161-169.
[2]	LIMA, et all (2006). Matemática do Ensino Médio, Volume 2. SBM, Rio de Janeiro.
[3]	OSONE, M. (February 2013) “Problemão Disfarçado de Probleminha”. São Paulo: Editora Segmento, Revista Calculo. pp. 20-26.
[4]	LAGARIAS, J. C. (2013) The 3x+1 Problem An Overview. Retrieved on August 20, 2013 from http://www.ams.org/bookstore/pspdf/mbk-78-prev.pdf.
[5]	CARVALHO, M. C. C. S. (2000) Padrões Numéricos e Sequências. São Paulo.
[6]	CHAVES, A. P. A. (2013) Equações Diofantinas Envolvendo Potências de Termos de Sequências Recorrentes. Brasilia: Universidade de Brasília (UNB). p. 1.
[7]	COUTINHO, S.C. Números Inteiros e Criptografia RSA. Rio de Janeiro, IMPA, 2011. p. 80
[8]	NETO, A. C. M. (2012) Tópicos de Matemática Elementar, Volume 5: Teoria dos Números. Rio de Janeiro: Number Theory. Vol. 5.
[9]	HEFEZ, M. (2011) Elementos de Aritmética. Rio de Janeiro: SBM. pp. 43-52.
[10]	SCHEINERMAN, E. R. (2006) Matemática Discreta. São Paulo. Câmara Brasileira do Livro.
[11]	LESIEUTRE,J (2007). On a Generalization of the Collatz Conjecture. Massachusetts. MIT (Massachusetts Institute of Technology).
[12]	CARNIELLY, W. Some Natural Generalizations Of The Collatz Problem. Applied Mathematics E-Notes, 15 (2015). < http://www.math.nthu.edu.tw/amen/>.
[13]	SINGH, S. (1998). O Último Teorema de Fermat. Rio de Janeiro. Editora Record.
[14]	SANTOS, O.O. (2016) Bases Numéricas Equações e Criptografia. São Paulo. All Print Editora.
[15]	STEWART, I. (2014) Os Maiores Problemas Matemáticos de Todos os Tempos. Rio de Janeiro: Zahar.
[16]	CASINI, A. et al. (2015) Um Problema di Convergerza de Tipo Collatz. Retrieved on June 3, from http://crf.uniroma2.it/wp-content/uploads/2010/04/Un-problema-di-convergenza-di-tipo-Collatz.pdf.

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Olinto de Oliveira Santos. (2018). Proving the Collatz Conjecture with Binaries Numbers. Pure and Applied Mathematics Journal, 7(5), 68-77. https://doi.org/10.11648/j.pamj.20180705.12

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Olinto de Oliveira Santos. Proving the Collatz Conjecture with Binaries Numbers. Pure Appl. Math. J. 2018, 7(5), 68-77. doi: 10.11648/j.pamj.20180705.12

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AMA Style

Olinto de Oliveira Santos. Proving the Collatz Conjecture with Binaries Numbers. Pure Appl Math J. 2018;7(5):68-77. doi: 10.11648/j.pamj.20180705.12

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@article{10.11648/j.pamj.20180705.12,
  author = {Olinto de Oliveira Santos},
  title = {Proving the Collatz Conjecture with Binaries Numbers},
  journal = {Pure and Applied Mathematics Journal},
  volume = {7},
  number = {5},
  pages = {68-77},
  doi = {10.11648/j.pamj.20180705.12},
  url = {https://doi.org/10.11648/j.pamj.20180705.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20180705.12},
  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.},
 year = {2018}
}

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TY  - JOUR
T1  - Proving the Collatz Conjecture with Binaries Numbers
AU  - Olinto de Oliveira Santos
Y1  - 2018/12/24
PY  - 2018
N1  - https://doi.org/10.11648/j.pamj.20180705.12
DO  - 10.11648/j.pamj.20180705.12
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 68
EP  - 77
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20180705.12
AB  - 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.
VL  - 7
IS  - 5
ER  -

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Author Information

Olinto de Oliveira Santos

CETEP- Territorial Center for Education Professional of the Coast of the Discovery, Eunápolis, Brazil

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APA Style

Olinto de Oliveira Santos. (2018). Proving the Collatz Conjecture with Binaries Numbers. Pure and Applied Mathematics Journal, 7(5), 68-77. https://doi.org/10.11648/j.pamj.20180705.12

Copy | Download

ACS Style

Olinto de Oliveira Santos. Proving the Collatz Conjecture with Binaries Numbers. Pure Appl. Math. J. 2018, 7(5), 68-77. doi: 10.11648/j.pamj.20180705.12

Copy | Download

AMA Style

Olinto de Oliveira Santos. Proving the Collatz Conjecture with Binaries Numbers. Pure Appl Math J. 2018;7(5):68-77. doi: 10.11648/j.pamj.20180705.12

Copy | Download

@article{10.11648/j.pamj.20180705.12,
  author = {Olinto de Oliveira Santos},
  title = {Proving the Collatz Conjecture with Binaries Numbers},
  journal = {Pure and Applied Mathematics Journal},
  volume = {7},
  number = {5},
  pages = {68-77},
  doi = {10.11648/j.pamj.20180705.12},
  url = {https://doi.org/10.11648/j.pamj.20180705.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20180705.12},
  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.},
 year = {2018}
}

Copy | Download

TY  - JOUR
T1  - Proving the Collatz Conjecture with Binaries Numbers
AU  - Olinto de Oliveira Santos
Y1  - 2018/12/24
PY  - 2018
N1  - https://doi.org/10.11648/j.pamj.20180705.12
DO  - 10.11648/j.pamj.20180705.12
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 68
EP  - 77
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20180705.12
AB  - 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.
VL  - 7
IS  - 5
ER  -

Copy | Download