Considerations on the Collatz Conjecture

Detalhes bibliográficos
Autor(a) principal: Loris, Audemir
Data de Publicação: 2024
Tipo de documento: preprint
Idioma: por
Título da fonte: SciELO Preprints
Texto Completo: https://preprints.scielo.org/index.php/scielo/preprint/view/7664
Resumo: Part of the scientific community has spent considerable time and resources to somehow validate Collatz's conjecture, countless efforts have achieved considerable progress in this direction, however this conjecture lacked definitive confirmation that choosing any odd number xi ∈ N∗, we will obtain x(i+1) = 3xi + 1, this being an even number x(i+1), divide it by the number two (successively) until another odd number ∈ N∗ is obtained, the process is repeated xn = 3x(n−1) + 1 and divisions by two until a number equal to 1 finally results.This work presents deductions, algorithms and equations that corroborate this proposition, supporting this perception and conclusion that Collatz's conjecture points to the final cycle 4 → 2 → 1.
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spelling Considerations on the Collatz ConjectureConsideraciones sobre la conjetura de CollatzConsiderações sabre a Conjectura de CollatzConjectura de CollatzDinâmica CaóticaCiclo LimiteOrbita periódicaPrincipio da indução matemáticaLinguagem Python e `R`Collatz conjectureChaotic DynamicsLimit CyclePeriodic orbitPrinciple of mathematical inductionPython and `R` languagePart of the scientific community has spent considerable time and resources to somehow validate Collatz's conjecture, countless efforts have achieved considerable progress in this direction, however this conjecture lacked definitive confirmation that choosing any odd number xi ∈ N∗, we will obtain x(i+1) = 3xi + 1, this being an even number x(i+1), divide it by the number two (successively) until another odd number ∈ N∗ is obtained, the process is repeated xn = 3x(n−1) + 1 and divisions by two until a number equal to 1 finally results.This work presents deductions, algorithms and equations that corroborate this proposition, supporting this perception and conclusion that Collatz's conjecture points to the final cycle 4 → 2 → 1.Parte de la comunidad científica ha dedicado un tiempo y recursos considerables a validar de alguna manera la conjetura de Collatz, innumerables esfuerzos han logrado avances considerables en esta dirección, sin embargo a esta conjetura le faltaba una confirmación definitiva de que eligiendo cualquier número impar xi ∈ N∗, obtendremos x(i+ 1) = 3xi + 1, siendo este un número par x(i+1), se divide por el número dos (sucesivamente) hasta obtener otro número impar ∈ N∗, se repite el proceso xn = 3x(n−1) + 1 y divisiones por dos hasta que finalmente resulte un número igual a 1.Este trabajo presenta deducciones, algoritmos y ecuaciones que corroboran esta proposición, apoyando esta percepción y conclusión de que la conjetura de Collatz apunta al ciclo final 4 → 2 → 1.Parte da comunidade cientifica tem dispendido considerável tempo e recursos para de alguma forma validar a conjectura de Collatz, inúmeros esforços tem logrado considerável avanço nesta direção, porém tal conjectura carecia de uma confirmação definitiva de que escolhido um número impar qualquer xi ∈ N∗, obteremos x(i+1) = 3xi + 1, sendo este um número x(i+1) par, divide-se o mesmo pelo número dois (sucessivamente) até obter-se outro número impar ∈ N∗ , repete-se o processo xn = 3x(n−1) + 1 e divisões por dois até que resulte finalmente um número igual 1.Neste trabalho apresentam-se deduções, algoritmos e equações que corroboram com tal proposição, embasam tal percepção e conclusão de que a conjectura de Collatz aponta para o ciclo final 4 → 2 → 1.SciELO PreprintsSciELO PreprintsSciELO Preprints2024-01-30info:eu-repo/semantics/preprintinfo:eu-repo/semantics/publishedVersionapplication/pdfhttps://preprints.scielo.org/index.php/scielo/preprint/view/766410.1590/SciELOPreprints.7664porhttps://preprints.scielo.org/index.php/scielo/article/view/7664/14926Copyright (c) 2024 Audemir Lorishttps://creativecommons.org/licenses/by/4.0info:eu-repo/semantics/openAccessLoris, Audemirreponame:SciELO Preprintsinstname:Scientific Electronic Library Online (SCIELO)instacron:SCI2023-12-08T14:04:35Zoai:ops.preprints.scielo.org:preprint/7664Servidor de preprintshttps://preprints.scielo.org/index.php/scieloONGhttps://preprints.scielo.org/index.php/scielo/oaiscielo.submission@scielo.orgopendoar:2023-12-08T14:04:35SciELO Preprints - Scientific Electronic Library Online (SCIELO)false
dc.title.none.fl_str_mv Considerations on the Collatz Conjecture
Consideraciones sobre la conjetura de Collatz
Considerações sabre a Conjectura de Collatz
title Considerations on the Collatz Conjecture
spellingShingle Considerations on the Collatz Conjecture
Loris, Audemir
Conjectura de Collatz
Dinâmica Caótica
Ciclo Limite
Orbita periódica
Principio da indução matemática
Linguagem Python e `R`
Collatz conjecture
Chaotic Dynamics
Limit Cycle
Periodic orbit
Principle of mathematical induction
Python and `R` language
title_short Considerations on the Collatz Conjecture
title_full Considerations on the Collatz Conjecture
title_fullStr Considerations on the Collatz Conjecture
title_full_unstemmed Considerations on the Collatz Conjecture
title_sort Considerations on the Collatz Conjecture
author Loris, Audemir
author_facet Loris, Audemir
author_role author
dc.contributor.author.fl_str_mv Loris, Audemir
dc.subject.por.fl_str_mv Conjectura de Collatz
Dinâmica Caótica
Ciclo Limite
Orbita periódica
Principio da indução matemática
Linguagem Python e `R`
Collatz conjecture
Chaotic Dynamics
Limit Cycle
Periodic orbit
Principle of mathematical induction
Python and `R` language
topic Conjectura de Collatz
Dinâmica Caótica
Ciclo Limite
Orbita periódica
Principio da indução matemática
Linguagem Python e `R`
Collatz conjecture
Chaotic Dynamics
Limit Cycle
Periodic orbit
Principle of mathematical induction
Python and `R` language
description Part of the scientific community has spent considerable time and resources to somehow validate Collatz's conjecture, countless efforts have achieved considerable progress in this direction, however this conjecture lacked definitive confirmation that choosing any odd number xi ∈ N∗, we will obtain x(i+1) = 3xi + 1, this being an even number x(i+1), divide it by the number two (successively) until another odd number ∈ N∗ is obtained, the process is repeated xn = 3x(n−1) + 1 and divisions by two until a number equal to 1 finally results.This work presents deductions, algorithms and equations that corroborate this proposition, supporting this perception and conclusion that Collatz's conjecture points to the final cycle 4 → 2 → 1.
publishDate 2024
dc.date.none.fl_str_mv 2024-01-30
dc.type.driver.fl_str_mv info:eu-repo/semantics/preprint
info:eu-repo/semantics/publishedVersion
format preprint
status_str publishedVersion
dc.identifier.uri.fl_str_mv https://preprints.scielo.org/index.php/scielo/preprint/view/7664
10.1590/SciELOPreprints.7664
url https://preprints.scielo.org/index.php/scielo/preprint/view/7664
identifier_str_mv 10.1590/SciELOPreprints.7664
dc.language.iso.fl_str_mv por
language por
dc.relation.none.fl_str_mv https://preprints.scielo.org/index.php/scielo/article/view/7664/14926
dc.rights.driver.fl_str_mv Copyright (c) 2024 Audemir Loris
https://creativecommons.org/licenses/by/4.0
info:eu-repo/semantics/openAccess
rights_invalid_str_mv Copyright (c) 2024 Audemir Loris
https://creativecommons.org/licenses/by/4.0
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv SciELO Preprints
SciELO Preprints
SciELO Preprints
publisher.none.fl_str_mv SciELO Preprints
SciELO Preprints
SciELO Preprints
dc.source.none.fl_str_mv reponame:SciELO Preprints
instname:Scientific Electronic Library Online (SCIELO)
instacron:SCI
instname_str Scientific Electronic Library Online (SCIELO)
instacron_str SCI
institution SCI
reponame_str SciELO Preprints
collection SciELO Preprints
repository.name.fl_str_mv SciELO Preprints - Scientific Electronic Library Online (SCIELO)
repository.mail.fl_str_mv scielo.submission@scielo.org
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