Representation of prime numbers on the complex plane

Representation of prime numbers on the complex plane | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Short Report Representation of prime numbers on the complex plane Levente Csóka This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-2869962/v4 This work is licensed under a CC BY 4.0 License Status: Posted Version 4 posted You are reading this latest preprint version Show more versions Abstract This study proposes a novel representation of prime numbers implying a quasiperiodic pattern within their distribution. Based on the result that any real number, and consequently, a prime number, can be represented in a complex plane using Euler’s notation; we investigated the quantities characterizing the sum of centres, which were observed as prime numbers. The numerical results indicate that prime numbers can be generated by rooting the real numbers, and their progression can be decomposed into periodic sequences. Within this framework, a harmonic prime function was identified, offering new insights into the structural intricacies of prime number sequences. The methodology employed advanced techniques to uncover the hidden periodicity underlying the quasiperiodic arithmetic progressions of primes, shedding light on the deeper patterns governing their distribution. Discrete Mathematics Prime numbers quasiperiodic pattern complex plane Euler's notation periodic sequences harmonic prime function Full Text Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 4 posted You are reading this latest preprint version Show more versions Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-2869962","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Short Report","associatedPublications":[],"authors":[{"id":282417457,"identity":"360c3b24-4770-443e-800b-3cfc25736cec","order_by":0,"name":"Levente Csóka","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAr0lEQVRIie3PIQ7CQBCF4dlM0pomaxfTM0xSieAqXYVFImmaDLLXqd2s7QEq2xu0DhIECwmWDg4xv5g884kB0LS/DJuphhIgT9tcRMS0VEOVKMoJO/iJ2C7wee4JLNowmT5uEzd6Hv1AsGsRyQwCQs4kwg+giFnaEmIDnzwTHCLmNxkBz/AihJiBiKRfWpdI4SJWaRy3ie1iWO9Mpb0287Lyfpt8Kt63lgNN0zTtW0+rXTLHK/aXmwAAAABJRU5ErkJggg==","orcid":"https://orcid.org/0000-0002-9687-7709","institution":"ELTE Eotvos Lorand University","correspondingAuthor":true,"prefix":"","firstName":"Levente","middleName":"","lastName":"Csóka","suffix":""}],"badges":[],"createdAt":"2023-04-27 18:54:36","currentVersionCode":4,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-2869962/v4","doiUrl":"https://doi.org/10.21203/rs.3.rs-2869962/v4","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":74098134,"identity":"a26c6f4b-492d-418f-aff0-4045cf7f8896","added_by":"auto","created_at":"2025-01-17 17:40:26","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":319161,"visible":true,"origin":"","legend":"","description":"","filename":"manuscriptRepresentationofprimenumbersinthecomplexplane.pdf","url":"https://assets-eu.researchsquare.com/files/rs-2869962/v4_covered_63bee1ee-9a02-4c25-99f1-24de652f3b7e.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"Representation of prime numbers on the complex plane","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"no fundings","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Prime numbers, quasiperiodic pattern, complex plane, Euler's notation, periodic sequences, harmonic prime function","lastPublishedDoi":"10.21203/rs.3.rs-2869962/v4","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-2869962/v4","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis study proposes a novel representation of prime numbers implying a quasiperiodic pattern within their distribution. 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