Structural Patterns of Goldbach Partition Numbers: A High-Precision Estimation Model Based on Prime Density

Structural Patterns of Goldbach Partition Numbers: A High-Precision Estimation Model Based on Prime Density | 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 Structural Patterns of Goldbach Partition Numbers: A High-Precision Estimation Model Based on Prime Density Zhao Shijian This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6510320/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract This research proposes a new approach to the Goldbach Conjecture based on the relationship between the partition numbers of even integers and interval prime density. Through in-depth analysis of even number decomposition properties, we discovered that even numbers can be classified into two categories: Type I (without effective prime factors) and Type II (with effective prime factors). This classification method makes the calculation of even number partition numbers more systematic. In our study, we defined the set X = {x|x = 2n - p, 3 ≤ p ≤ p_m ≤ n}, where p is an odd prime, and through rigorous mathematical proof and large-scale numerical validation, confirmed that there exists a constant ratio (approximately 1.32α) between the prime density in set X and the prime density in the interval [ n , 2 n ]. Based on this discovery, we established an accurate partition number estimation model: G (2 n ) = c 1 α nf 1 f 2 , where c 1 ≈ 1.32 is the Goldbach constant, α is the even number factor coefficient, and f 1 and f 2 are the prime densities in intervals [3, n ] and [ n , 2 n ] respectively. This research conducted comprehensive validation within the range of 1 billion and representative sampling in the 10-100 billion range, showing that the estimation model has high accuracy, with relative errors not exceeding 0.2% for even numbers above 100 million. More importantly, we proved that the G (2 n ) values for Type I even numbers increase monotonically as n increases, and that partitioning values for Type I even numbers are typically smaller than those for Type II even numbers, thus providing a new path for proving the Goldbach Conjecture. Mathematics Subject Classification (2020): 11P32, 11N05, 11Y35, 11Y16 Goldbach conjecture Partition numbers Prime density Even number classification Number theory Full Text Additional Declarations No competing interests reported. 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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-6510320","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":448604033,"identity":"9cfadbfd-fbfc-4d33-9d27-24fa4834ea0c","order_by":0,"name":"Zhao Shijian","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABE0lEQVRIiWNgGAWjYDACCSBmbJBgZgNSDxIqauRAggceEKmF2eDDmWPGYC0JhLWAmWySM9uYE8FsfFrkZzc/fPhzhwU7H/vZA9I8bGzp88MOPwTaYien24BdC+OcY8bGvGeADuPJSzDm4ZHJ3Xg7zQCoJdnY7AB2LcwSCWbSjG0gv+QYJPNIsOVunJ0A0nIgcRsOLWwS6d8kf4K08L8xOMxjwJxuODv9A14tPBI5ZhK8IC0SOYaNMxKYE+Slc/DbIiGRU2wM0fLGmOHDgWOGG6RzCg4kGOD2i/yM9I0Pf7bVJcv355j/SPxXIy8/O33zhw8VdnK4tMBAMpxlAFZpgF85CNgh7G0grHoUjIJRMApGFgAAiOZaLusaOMgAAAAASUVORK5CYII=","orcid":"","institution":"","correspondingAuthor":true,"prefix":"","firstName":"Zhao","middleName":"","lastName":"Shijian","suffix":""}],"badges":[],"createdAt":"2025-04-23 08:08:21","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6510320/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6510320/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":81607551,"identity":"56bc35ee-32fc-45b0-90e0-a154fed61e74","added_by":"auto","created_at":"2025-04-29 06:13:07","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1962452,"visible":true,"origin":"","legend":"","description":"","filename":"GoldbachLaTeXSource.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6510320/v1_covered_541c6848-a20e-4650-9936-836201e56194.pdf"},{"id":81606710,"identity":"25f4ccb7-ed6f-43cb-b1c4-7d3d299723ac","added_by":"auto","created_at":"2025-04-29 06:05:04","extension":"zip","order_by":0,"title":"","display":"","copyAsset":false,"role":"supplement","size":181049,"visible":true,"origin":"","legend":"","description":"","filename":"ASupplementaryMaterials.zip","url":"https://assets-eu.researchsquare.com/files/rs-6510320/v1/fdb4f9450f3348d932480b49.zip"}],"financialInterests":"No competing interests reported.","formattedTitle":"Structural Patterns of Goldbach Partition Numbers: A High-Precision Estimation Model Based on Prime Density","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":"Goldbach conjecture, Partition numbers, Prime density, Even number classification, Number theory","lastPublishedDoi":"10.21203/rs.3.rs-6510320/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6510320/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis research proposes a new approach to the Goldbach Conjecture based on the relationship between the partition numbers of even integers and interval prime density. Through in-depth analysis of even number decomposition properties, we discovered that even numbers can be classified into two categories: Type I (without effective prime factors) and Type II (with effective prime factors). This classification method makes the calculation of even number partition numbers more systematic.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eIn our study, we defined the set X = {x|x = 2n - p, 3 ≤ p ≤ p_m ≤ n}, where \u003cem\u003ep\u003c/em\u003e is an odd prime, and through rigorous mathematical proof and large-scale numerical validation, confirmed that there exists a constant ratio (approximately 1.32α) between the prime density in set \u003cem\u003eX\u003c/em\u003e and the prime density in the interval [\u003cem\u003en\u003c/em\u003e, 2\u003cem\u003en\u003c/em\u003e]. Based on this discovery, we established an accurate partition number estimation model: \u003cem\u003eG\u003c/em\u003e(2\u003cem\u003en\u003c/em\u003e) = c\u003csub\u003e1\u003c/sub\u003eα nf\u003csub\u003e1\u003c/sub\u003ef\u003csub\u003e2\u003c/sub\u003e, where c\u003csub\u003e1\u003c/sub\u003e ≈ 1.32 is the Goldbach constant, \u003cem\u003eα\u003c/em\u003e is the even number factor coefficient, and f\u003csub\u003e1\u003c/sub\u003e and f\u003csub\u003e2\u003c/sub\u003e are the prime densities in intervals [3, \u003cem\u003en\u003c/em\u003e] and [\u003cem\u003en\u003c/em\u003e, 2\u003cem\u003en\u003c/em\u003e] respectively.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eThis research conducted comprehensive validation within the range of 1 billion and representative sampling in the 10-100 billion range, showing that the estimation model has high accuracy, with relative errors not exceeding 0.2% for even numbers above 100 million. More importantly, we proved that the \u003cem\u003eG\u003c/em\u003e(2\u003cem\u003en\u003c/em\u003e) values for Type I even numbers increase monotonically as \u003cem\u003en\u003c/em\u003e increases, and that partitioning values for Type I even numbers are typically smaller than those for Type II even numbers, thus providing a new path for proving the Goldbach Conjecture.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMathematics Subject Classification (2020):\u003c/strong\u003e 11P32, 11N05, 11Y35, 11Y16\u003c/p\u003e","manuscriptTitle":"Structural Patterns of Goldbach Partition Numbers: A High-Precision Estimation Model Based on Prime Density","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-04-29 06:04:59","doi":"10.21203/rs.3.rs-6510320/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":"d059b646-5bfc-4938-8bdf-07f97e0e5751","owner":[],"postedDate":"April 29th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2025-04-29T06:04:59+00:00","versionOfRecord":[],"versionCreatedAt":"2025-04-29 06:04:59","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-6510320","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-6510320","identity":"rs-6510320","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}