{"paper_id":"10f1aa87-248a-47af-9fe8-e96a75c8bad7","body_text":"Twin Prime Conjecture Proof | 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Cite Share Download PDF Research Article Twin Prime Conjecture Proof Daniel Thompson This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-2619857/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 The Twin Prime conjecture first proposed by French mathematician Alphonse de Polignac in the year 1846 remains unproven until now. This paper sets out to prove the Twin Prime conjecture by creating a system of equations and finding both a y element of ℕ, (y ϵ ℕ) and a y element of k, (y ϵ k) to find the twin prime values and then two further values where each will differ by an exact amount. From this proof the method demonstrates that the elements of these phenomena prove that there will be an infinite number of twin primes that differ by two. MSC: 11-xx Number theory Twin Prime conjecture and Polignac’s conjecture 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. 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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-2619857\",\"acceptedTermsAndConditions\":true,\"allowDirectSubmit\":true,\"archivedVersions\":[],\"articleType\":\"Research Article\",\"associatedPublications\":[],\"authors\":[{\"id\":271650053,\"identity\":\"7b72c78e-9596-4940-b16c-ca4357f14820\",\"order_by\":0,\"name\":\"Daniel Thompson\",\"email\":\"data:image/png;base64,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\",\"orcid\":\"\",\"institution\":\"\",\"correspondingAuthor\":true,\"prefix\":\"\",\"firstName\":\"Daniel\",\"middleName\":\"\",\"lastName\":\"Thompson\",\"suffix\":\"\"}],\"badges\":[],\"createdAt\":\"2023-02-23 09:44:37\",\"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-2619857/v3\",\"doiUrl\":\"https://doi.org/10.21203/rs.3.rs-2619857/v3\",\"draftVersion\":[],\"editorialEvents\":[],\"editorialNote\":\"\",\"failedWorkflow\":false,\"files\":[{\"id\":52442760,\"identity\":\"46e2db7c-4e95-4b8a-a588-1e7b7c7857c6\",\"added_by\":\"auto\",\"created_at\":\"2024-03-11 17:30:26\",\"extension\":\"pdf\",\"order_by\":1,\"title\":\"\",\"display\":\"\",\"copyAsset\":false,\"role\":\"manuscript-pdf\",\"size\":195040,\"visible\":true,\"origin\":\"\",\"legend\":\"\",\"description\":\"\",\"filename\":\"TwimPrimeConjectureProof.pdf\",\"url\":\"https://assets-eu.researchsquare.com/files/rs-2619857/v3_covered_d7f092d8-fb42-4fc1-8756-e72a9f2514ea.pdf\"}],\"financialInterests\":\"The authors declare no competing interests.\",\"formattedTitle\":\"Twin Prime Conjecture Proof\",\"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\":\"info@researchsquare.com\",\"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\":\"Number theory, Twin Prime conjecture and Polignac’s conjecture\",\"lastPublishedDoi\":\"10.21203/rs.3.rs-2619857/v3\",\"lastPublishedDoiUrl\":\"https://doi.org/10.21203/rs.3.rs-2619857/v3\",\"license\":{\"name\":\"CC BY 4.0\",\"url\":\"https://creativecommons.org/licenses/by/4.0/\"},\"manuscriptAbstract\":\"\\u003cp\\u003eThe Twin Prime conjecture first proposed by French mathematician Alphonse de Polignac in the year 1846 remains unproven until now. 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