Structure in prime gaps

Structure in prime gaps | 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a Preprint Cite Share Download PDF Research Article Structure in prime gaps KAJANI KAUNDA This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4058806/v3 This work is licensed under a CC BY 4.0 License Status: Posted Version 3 posted You are reading this latest preprint version Show more versions Abstract We define a Cayley Table T from the structure ( J ,+) based on a subset J of Z containing prime numbers and their additive inverses, which we then use as a model of gaps between primes of the form p α − 3. Using definitions of the relationships between primes and their gaps derived from T , we prove the existence of infinitely many pairs of primes, ( p n , p n+m ), such that ( p n+m − p n ) = ( p α − 3) where n , α ≥ 3 and m ≥ 1 and p n is the n th prime. Finally, we use this result to show the existence of infinitely many pairs of prime numbers with a gap of 2. MSC Classification: 11N05, 11B05 Pure Mathematics Prime Numbers Twin Prime Conjecture Prime Gaps Number Theory Cayley Tables Set Theory Analytic Number Theory Full Text Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 3 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-4058806","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":319955735,"identity":"6ee7e350-5081-4ed8-b281-7bfe10f97926","order_by":0,"name":"KAJANI KAUNDA","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA/UlEQVRIiWNgGAWjYBACCQaGBAYGgwQQ24DhA5BkYydFC+MMkBZmwlpAAKKFmQdEEdIi2X7g4eeCgjR5g9vNGz/b/Nomz8fMwPjhYw5uLdI8CcnSMwxyDDfcOVYsndt327CNmYFZcuY23FrkGBISpHkMKhg33MgxkM7tuc0I1MLGzItPC/+D5N9ALfZALca/LXtu2xPUIi2RkAa0JScRqMVMmuHH7USCWiRnPEiz5jFIS555I63MsrfhdnIbM2MzXr9InM9Jvs3zJ9m270by5hs//ty2nd/efPDDRzxaGBh4EhBsxjYw2YBPPRCwH0Di/CGgeBSMglEwCkYkAAAIz1BIweYc6AAAAABJRU5ErkJggg==","orcid":"https://orcid.org/0000-0003-3737-0032","institution":"","correspondingAuthor":true,"prefix":"","firstName":"KAJANI","middleName":"","lastName":"KAUNDA","suffix":""}],"badges":[],"createdAt":"2024-03-09 17:47:31","currentVersionCode":3,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-4058806/v3","doiUrl":"https://doi.org/10.21203/rs.3.rs-4058806/v3","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":60422507,"identity":"08559ebd-6cd1-40e0-8469-166e5abdfca9","added_by":"auto","created_at":"2024-07-16 15:02:29","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":384316,"visible":true,"origin":"","legend":"","description":"","filename":"StructureinPrimeGaps.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4058806/v3_covered_b06f8d8e-6158-4ec6-853d-f0cf7035ed64.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"Structure in prime gaps","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","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, Twin Prime Conjecture, Prime Gaps, Number Theory, Cayley Tables, Set Theory, Analytic Number Theory","lastPublishedDoi":"10.21203/rs.3.rs-4058806/v3","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4058806/v3","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe define a Cayley Table \u003cem\u003eT\u003c/em\u003e from the structure (\u003cem\u003eJ\u003c/em\u003e,+) based on a subset \u003cem\u003eJ\u003c/em\u003e of \u003cem\u003eZ\u003c/em\u003e containing prime numbers and their additive inverses, which we then use as a model of gaps between primes of the form \u003cem\u003ep\u003c/em\u003e\u003csub\u003eα\u003c/sub\u003e − 3. Using definitions of the relationships between primes and their gaps derived from \u003cem\u003eT\u003c/em\u003e, we prove the existence of infinitely many pairs of primes, (\u003cem\u003ep\u003c/em\u003e\u003csub\u003en\u003c/sub\u003e, \u003cem\u003ep\u003c/em\u003e\u003csub\u003en+m\u003c/sub\u003e), such that (\u003cem\u003ep\u003c/em\u003e\u003csub\u003en+m\u003c/sub\u003e − \u003cem\u003ep\u003c/em\u003e\u003csub\u003en\u003c/sub\u003e) = (\u003cem\u003ep\u003c/em\u003e\u003csub\u003eα\u003c/sub\u003e − 3) where \u003cem\u003en\u003c/em\u003e, α ≥ 3 and \u003cem\u003em\u003c/em\u003e ≥ 1 and \u003cem\u003ep\u003c/em\u003e\u003csub\u003en\u003c/sub\u003e is the \u003cem\u003en\u003c/em\u003e\u003csup\u003eth\u003c/sup\u003e prime. Finally, we use this result to show the existence of infinitely many pairs of prime numbers with a gap of 2.\u003c/p\u003e\n\u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n\u003cp\u003eMSC Classification: 11N05, 11B05\u003c/p\u003e","manuscriptTitle":"Structure in prime gaps","msid":"","msnumber":"","nonDraftVersions":[{"code":3,"date":"2024-07-16 14:46:21","doi":"10.21203/rs.3.rs-4058806/v3","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","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}},{"code":2,"date":"2024-06-27 21:13:41","doi":"10.21203/rs.3.rs-4058806/v2","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","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}},{"code":1,"date":"2024-03-11 03:12:16","doi":"10.21203/rs.3.rs-4058806/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","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}}],"origin":"","ownerIdentity":"ce46fc18-aeaa-4b9c-8d50-09e4ec19dfe3","owner":[],"postedDate":"July 16th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":33834443,"name":"Pure Mathematics"}],"tags":[],"updatedAt":"2024-03-11T03:12:16+00:00","versionOfRecord":[],"versionCreatedAt":"2024-07-16 14:46:21","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v3","identity":"rs-4058806","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-4058806","identity":"rs-4058806","version":["v3"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}