A Novel Prime Number Identity Bridging the Collatz Conjecture and Additive Prime Structures

A Novel Prime Number Identity Bridging the Collatz Conjecture and Additive Prime Structures | 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 Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed.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. 15 May 2025 V1 Latest version Share on A Novel Prime Number Identity Bridging the Collatz Conjecture and Additive Prime Structures Author : Budee U Zaman 0009-0004-2896-3586 [email protected] Authors Info & Affiliations https://doi.org/10.22541/au.174733684.46942320/v1 566 views 139 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract This paper presents a novel identity that establishes a surprising link between prime number theory and the Collatz Conjecture, two foundational yet independently explored areas of number theory. The identity, n i=1 pi − n−1 i=2 gi(n − i) + 2 = 3n + 1, [9] where pi denotes the ith prime number, offers a structured summation that yields a linear expression in n, namely 3n + 1. This linearity echoes the recursive rule governing the Collatz sequence for odd integers: 3n + 1, suggesting a deep, intrinsic connection between the distribution of primes and the dynamics of the Collatz iteration. The study explores this identity analytically and numerically, providing insights into its validity, scope, and potential implications. The structure and behavior of the difference terms (pi+1 − pi)(n − i) are also analyzed, highlighting a hidden regularity in prime gaps when viewed through the lens of this summation. The result opens new avenues for reinterpreting prime behavior in discrete systems and lays foundational groundwork for a possible unification of discrete dynamical processes and prime arithmetic. Supplementary Material File (a_novel_prime_number_identity_bridging_the_collatz_conjecture_and_additive_prime_structures (6).pdf) Download 267.57 KB Information & Authors Information Version history V1 Version 1 15 May 2025 Copyright This work is licensed under a Creative Commons Attribution 4.0 International License Keywords collatz integers natural numbers prime gaps prime number Authors Affiliations Budee U Zaman 0009-0004-2896-3586 [email protected] View all articles by this author Funding Information Ford Foundation Metrics & Citations Metrics Article Usage 566 views 139 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Budee U Zaman. A Novel Prime Number Identity Bridging the Collatz Conjecture and Additive Prime Structures. Authorea . 15 May 2025. DOI: https://doi.org/10.22541/au.174733684.46942320/v1 If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. 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