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.
Copyright © 2018 Authors retain the copyright of this article.
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FILHO, A. C. S. (2010) “Formulações Alternativas para Conjectura de Collatz”. XI Encontro de Pesquisadores. pp. 161-169.
LIMA, et all (2006). Matemática do Ensino Médio, Volume 2. SBM, Rio de Janeiro.
OSONE, M. (February 2013) “Problemão Disfarçado de Probleminha”. São Paulo: Editora Segmento, Revista Calculo. pp. 20-26.
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.
CARVALHO, M. C. C. S. (2000) Padrões Numéricos e Sequências. São Paulo.
CHAVES, A. P. A. (2013) Equações Diofantinas Envolvendo Potências de Termos de Sequências Recorrentes. Brasilia: Universidade de Brasília (UNB). p. 1.
COUTINHO, S.C. Números Inteiros e Criptografia RSA. Rio de Janeiro, IMPA, 2011. p. 80
NETO, A. C. M. (2012) Tópicos de Matemática Elementar, Volume 5: Teoria dos Números. Rio de Janeiro: Number Theory. Vol. 5.
HEFEZ, M. (2011) Elementos de Aritmética. Rio de Janeiro: SBM. pp. 43-52.
SCHEINERMAN, E. R. (2006) Matemática Discreta. São Paulo. Câmara Brasileira do Livro.
LESIEUTRE,J (2007). On a Generalization of the Collatz Conjecture. Massachusetts. MIT (Massachusetts Institute of Technology).
CARNIELLY, W. Some Natural Generalizations Of The Collatz Problem. Applied Mathematics E-Notes, 15 (2015). < http://www.math.nthu.edu.tw/amen/>.
SINGH, S. (1998). O Último Teorema de Fermat. Rio de Janeiro. Editora Record.
SANTOS, O.O. (2016) Bases Numéricas Equações e Criptografia. São Paulo. All Print Editora.
STEWART, I. (2014) Os Maiores Problemas Matemáticos de Todos os Tempos. Rio de Janeiro: Zahar.
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.