{"id":"https://openalex.org/W7160417401","doi":"https://doi.org/10.1109/access.2026.3690991","title":"Efficient Approximation of Goldbach Partition Function Using Machine Learning Models","display_name":"Efficient Approximation of Goldbach Partition Function Using Machine Learning Models","publication_year":2026,"publication_date":"2026-01-01","ids":{"openalex":"https://openalex.org/W7160417401","doi":"https://doi.org/10.1109/access.2026.3690991"},"language":"en","primary_location":{"id":"doi:10.1109/access.2026.3690991","is_oa":true,"landing_page_url":"https://doi.org/10.1109/access.2026.3690991","pdf_url":null,"source":{"id":"https://openalex.org/S2485537415","display_name":"IEEE Access","issn_l":"2169-3536","issn":["2169-3536"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Access","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://doi.org/10.1109/access.2026.3690991","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5135459771","display_name":"Cheng-Zhe Wu","orcid":"https://orcid.org/0009-0000-1559-978X"},"institutions":[{"id":"https://openalex.org/I192703390","display_name":"Ming Chuan University","ror":"https://ror.org/02pgvzy25","country_code":"TW","type":"education","lineage":["https://openalex.org/I192703390"]}],"countries":["TW"],"is_corresponding":false,"raw_author_name":"Cheng-Zhe Wu","raw_affiliation_strings":["Department of Computer Science and Information Engineering, Ming Chuan University, Taoyuan, Taiwan"],"raw_orcid":"https://orcid.org/0009-0000-1559-978X","affiliations":[{"raw_affiliation_string":"Department of Computer Science and Information Engineering, Ming Chuan University, Taoyuan, Taiwan","institution_ids":["https://openalex.org/I192703390"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5047522660","display_name":"Shyong Jian Shyu","orcid":"https://orcid.org/0000-0003-1372-7076"},"institutions":[{"id":"https://openalex.org/I192703390","display_name":"Ming Chuan University","ror":"https://ror.org/02pgvzy25","country_code":"TW","type":"education","lineage":["https://openalex.org/I192703390"]}],"countries":["TW"],"is_corresponding":false,"raw_author_name":"Shyong Jian Shyu","raw_affiliation_strings":["Department of Computer Science and Information Engineering, Ming Chuan University, Taoyuan, Taiwan"],"raw_orcid":"https://orcid.org/0000-0003-1372-7076","affiliations":[{"raw_affiliation_string":"Department of Computer Science and Information Engineering, Ming Chuan University, Taoyuan, Taiwan","institution_ids":["https://openalex.org/I192703390"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":{"value":1850,"currency":"USD","value_usd":1850},"apc_paid":{"value":1850,"currency":"USD","value_usd":1850},"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.73963376,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"14","issue":null,"first_page":"83095","last_page":"83108"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11166","display_name":"Analytic Number Theory Research","score":0.38749998807907104,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11166","display_name":"Analytic Number Theory Research","score":0.38749998807907104,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11716","display_name":"Random Matrices and Applications","score":0.12099999934434891,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T13720","display_name":"Benford\u2019s Law and Fraud Detection","score":0.04529999941587448,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/function-approximation","display_name":"Function approximation","score":0.5238999724388123},{"id":"https://openalex.org/keywords/approximation-algorithm","display_name":"Approximation algorithm","score":0.4684999883174896},{"id":"https://openalex.org/keywords/partition","display_name":"Partition (number theory)","score":0.45080000162124634},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.4498000144958496},{"id":"https://openalex.org/keywords/partition-function","display_name":"Partition function (quantum field theory)","score":0.4115999937057495},{"id":"https://openalex.org/keywords/goldbachs-conjecture","display_name":"Goldbach's conjecture","score":0.4065000116825104}],"concepts":[{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.6252999901771545},{"id":"https://openalex.org/C91873725","wikidata":"https://www.wikidata.org/wiki/Q3445816","display_name":"Function approximation","level":3,"score":0.5238999724388123},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.5128999948501587},{"id":"https://openalex.org/C148764684","wikidata":"https://www.wikidata.org/wiki/Q621751","display_name":"Approximation algorithm","level":2,"score":0.4684999883174896},{"id":"https://openalex.org/C42812","wikidata":"https://www.wikidata.org/wiki/Q1082910","display_name":"Partition (number theory)","level":2,"score":0.45080000162124634},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4498000144958496},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.448199987411499},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.41350001096725464},{"id":"https://openalex.org/C2778401447","wikidata":"https://www.wikidata.org/wiki/Q7140637","display_name":"Partition function (quantum field theory)","level":2,"score":0.4115999937057495},{"id":"https://openalex.org/C143732363","wikidata":"https://www.wikidata.org/wiki/Q485520","display_name":"Goldbach's conjecture","level":3,"score":0.4065000116825104},{"id":"https://openalex.org/C50292564","wikidata":"https://www.wikidata.org/wiki/Q2462783","display_name":"Computational learning theory","level":3,"score":0.34790000319480896},{"id":"https://openalex.org/C22367795","wikidata":"https://www.wikidata.org/wiki/Q7625208","display_name":"Structured prediction","level":2,"score":0.3433000147342682},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3230000138282776},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.313400000333786},{"id":"https://openalex.org/C51632099","wikidata":"https://www.wikidata.org/wiki/Q3985153","display_name":"Training set","level":2,"score":0.304500013589859},{"id":"https://openalex.org/C145242015","wikidata":"https://www.wikidata.org/wiki/Q774123","display_name":"Approximation theory","level":2,"score":0.2987000048160553},{"id":"https://openalex.org/C108583219","wikidata":"https://www.wikidata.org/wiki/Q197536","display_name":"Deep learning","level":2,"score":0.259799987077713}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1109/access.2026.3690991","is_oa":true,"landing_page_url":"https://doi.org/10.1109/access.2026.3690991","pdf_url":null,"source":{"id":"https://openalex.org/S2485537415","display_name":"IEEE Access","issn_l":"2169-3536","issn":["2169-3536"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Access","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:a3cff7c68ce64dd2a94a7f3dded19b2f","is_oa":true,"landing_page_url":"https://doaj.org/article/a3cff7c68ce64dd2a94a7f3dded19b2f","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"IEEE Access, Vol 14, Pp 83095-83108 (2026)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.1109/access.2026.3690991","is_oa":true,"landing_page_url":"https://doi.org/10.1109/access.2026.3690991","pdf_url":null,"source":{"id":"https://openalex.org/S2485537415","display_name":"IEEE Access","issn_l":"2169-3536","issn":["2169-3536"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Access","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1820022171","display_name":null,"funder_award_id":"114-2221-E-130-006","funder_id":"https://openalex.org/F4320331164","funder_display_name":"National Science and Technology Council"},{"id":"https://openalex.org/G299609662","display_name":null,"funder_award_id":"113-2221-E-130-008","funder_id":"https://openalex.org/F4320331164","funder_display_name":"National Science and Technology Council"}],"funders":[{"id":"https://openalex.org/F4320331164","display_name":"National Science and Technology Council","ror":"https://ror.org/00wnb9798"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Goldbach\u2019s":[0],"conjecture,":[1],"still":[2],"unproven,":[3],"posits":[4],"that":[5,68,107],"every":[6],"even":[7,36],"integer":[8,37],"greater":[9],"than":[10],"2":[11],"can":[12],"be":[13],"expressed":[14],"as":[15,76],"the":[16,29],"sum":[17],"of":[18,31],"two":[19],"primes.":[20],"The":[21],"Goldbach":[22],"partition":[23],"function":[24,72],"<italic":[25,38,57],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[26,39,58,74],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">G(n)</i>":[27],"counts":[28],"number":[30],"such":[32],"prime":[33],"pairs":[34],"for":[35],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">n</i>":[40],">":[41],"2.":[42],"While":[43],"several":[44],"analytic":[45],"approximations":[46],"exist,":[47],"recent":[48],"studies":[49],"have":[50],"explored":[51],"machine":[52],"learning":[53],"approaches":[54],"to":[55],"estimate":[56],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">G(n)</i>.":[59],"Building":[60],"on":[61],"prior":[62],"architectures,":[63],"we":[64],"propose":[65],"refined":[66],"models":[67,109],"integrate":[69],"Euler\u2019s":[70],"totient":[71],"\u03c6<italic":[73],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">(n)</i>":[75],"a":[77],"primary":[78],"feature,":[79],"combined":[80],"with":[81,92],"nonlinear":[82],"logarithmic-exponential":[83],"transformations,":[84],"data":[85],"exchange,":[86],"residual":[87],"learning,":[88],"and":[89,101,120],"deeper":[90],"layers":[91],"increasing":[93],"unit":[94],"sizes.":[95],"These":[96],"innovations":[97],"enable":[98],"more":[99],"accurate":[100],"stable":[102],"approximations.":[103],"Experimental":[104],"results":[105],"confirm":[106],"our":[108],"consistently":[110],"outperform":[111],"existing":[112],"ones":[113],"by":[114],"achieving":[115],"lower":[116],"mean":[117],"squared":[118],"errors":[119],"Error":[121],"rates.":[122],"Our":[123],"best-performing":[124],"model":[125],"not":[126],"only":[127],"delivers":[128],"reliable":[129],"accuracy":[130],"but":[131],"also":[132],"demonstrates":[133],"strong":[134],"robustness":[135],"across":[136],"various":[137],"test":[138],"ranges.":[139]},"counts_by_year":[],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2026-05-07T00:00:00"}