Monotonicity and Convergence in the Collatz Conjecture: the Tendency of Integers to Reach 1     

Monotonicity and Convergence in the Collatz Conjecture: the Tendency of Integers to Reach 1 | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 24 March 2025 V5 Latest version Share on Monotonicity and Convergence in the Collatz Conjecture: the Tendency of Integers to Reach 1 Author : Guillermo Wells Abascal 0009-0009-1180-4147 [email protected] Authors Info & Affiliations https://doi.org/10.22541/au.172228097.77129028/v5 629 views 256 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract The Collatz conjecture declares that every positive integer will eventually reach 1 when subjected to a simple iterative process: if the number is even, it is divided by 2, and if it is odd, it is multiplied by 3 and then increased by 1. Despite the straightforward nature of these rules, a general proof of the conjecture remains elusive. For the above, this study introduces an alternative interpretation of the conjecture. This approach involves multiplying an odd integer N 1 by 3 and subsequently adding the largest power-of-2 factor within N 1 . Repeated iterations of this alternative process show that any initial odd integer N 1 will eventually convert into a power of 2, leading the sequence towards convergence. The behavior of the sequence was studied by representing integers as a power of 2 multiplied by an odd component. Using this representation under the modified rules, we developed a structured proof framework that demonstrates the consistent reduction of the odd component’s relative value after each iteration, the accelerated increase of the power-of-2 factor’s relative value, and the absence of any divergent cycles or alternative behaviors. This analysis provides insights into the mechanics of convergence in the Collatz sequence and proposes a new perspective for understanding the conjecture’s underlying dynamics. Supplementary Material File (authorea.pdf) Download 692.30 KB Information & Authors Information Version history V1 Version 1 29 July 2024 V2 Version 2 20 August 2024 V3 Version 3 09 September 2024 V4 Version 4 18 December 2024 V5 Version 5 24 March 2025 Copyright This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License Keywords algebra collatz conjecture maths number theory Authors Affiliations Guillermo Wells Abascal 0009-0009-1180-4147 [email protected] View all articles by this author Metrics & Citations Metrics Article Usage 629 views 256 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Guillermo Wells Abascal. Monotonicity and Convergence in the Collatz Conjecture: the Tendency of Integers to Reach 1 . Authorea . 24 March 2025. DOI: https://doi.org/10.22541/au.172228097.77129028/v5 If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download. For more information or tips please see 'Downloading to a citation manager' in the Help menu . Format Please select one from the list RIS (ProCite, Reference Manager) EndNote BibTex Medlars RefWorks Direct import Tips for downloading citations document.getElementById('citMgrHelpLink').addEventListener('click', function() { popupHelp(this.href); return false; }); $(".js__slcInclude").on("change", function(e){ if ($(this).val() == 'refworks') $('#direct').prop("checked", false); $('#direct').prop("disabled", ($(this).val() == 'refworks')); }); Cited by Eyob Solomon Getachew, Unfolding the Collatz Tree: An Indirect Structural Proof of the Collatz Conjecture, Research in Mathematics, 12 , 1, (2025). https://doi.org/10.1080/27684830.2025.2542052 Crossref Loading... View Options View options PDF View PDF Figures Tables Media Share Share Share article link Copy Link Copied! Copying failed. 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