{"id":"https://openalex.org/W2083393265","doi":"https://doi.org/10.2478/v10037-012-0020-2","title":"Extended Euclidean Algorithm and CRT Algorithm","display_name":"Extended Euclidean Algorithm and CRT Algorithm","publication_year":2012,"publication_date":"2012-12-01","ids":{"openalex":"https://openalex.org/W2083393265","doi":"https://doi.org/10.2478/v10037-012-0020-2","mag":"2083393265"},"language":"en","primary_location":{"id":"doi:10.2478/v10037-012-0020-2","is_oa":true,"landing_page_url":"https://doi.org/10.2478/v10037-012-0020-2","pdf_url":"https://sciendo.com/pdf/10.2478/v10037-012-0020-2","source":{"id":"https://openalex.org/S179241345","display_name":"Formalized Mathematics","issn_l":"1426-2630","issn":["1426-2630","1898-9934"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320322","host_organization_name":"De Gruyter Open","host_organization_lineage":["https://openalex.org/P4310320322","https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter Open","De Gruyter"],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formalized Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://sciendo.com/pdf/10.2478/v10037-012-0020-2","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5048322538","display_name":"Hiroyuki Okazaki","orcid":"https://orcid.org/0009-0000-1022-361X"},"institutions":[{"id":"https://openalex.org/I137975476","display_name":"Shinshu University","ror":"https://ror.org/0244rem06","country_code":"JP","type":"education","lineage":["https://openalex.org/I137975476"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Hiroyuki Okazaki","raw_affiliation_strings":["[Shinshu Univ., Nagano, Japan]"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"[Shinshu Univ., Nagano, Japan]","institution_ids":["https://openalex.org/I137975476"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5007677589","display_name":"Yosiki Aoki","orcid":null},"institutions":[{"id":"https://openalex.org/I137975476","display_name":"Shinshu University","ror":"https://ror.org/0244rem06","country_code":"JP","type":"education","lineage":["https://openalex.org/I137975476"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Yosiki Aoki","raw_affiliation_strings":["[Shinshu Univ., Nagano, Japan]"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"[Shinshu Univ., Nagano, Japan]","institution_ids":["https://openalex.org/I137975476"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5055549852","display_name":"Yasunari Shidama","orcid":"https://orcid.org/0000-0001-5634-0838"},"institutions":[{"id":"https://openalex.org/I137975476","display_name":"Shinshu University","ror":"https://ror.org/0244rem06","country_code":"JP","type":"education","lineage":["https://openalex.org/I137975476"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Yasunari Shidama","raw_affiliation_strings":["[Shinshu Univ., Nagano, Japan]"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"[Shinshu Univ., Nagano, Japan]","institution_ids":["https://openalex.org/I137975476"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I137975476"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":{"value":0.12553537,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"20","issue":"2","first_page":"175","last_page":"179"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12504","display_name":"Mathematics and Applications","score":0.9136999845504761,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/T12504","display_name":"Mathematics and Applications","score":0.9136999845504761,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/euclidean-algorithm","display_name":"Euclidean algorithm","score":0.9368791580200195},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.8178026676177979},{"id":"https://openalex.org/keywords/euclidean-geometry","display_name":"Euclidean geometry","score":0.6930805444717407},{"id":"https://openalex.org/keywords/chinese-remainder-theorem","display_name":"Chinese remainder theorem","score":0.63099604845047},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6220661401748657},{"id":"https://openalex.org/keywords/euclidean-domain","display_name":"Euclidean domain","score":0.6146183013916016},{"id":"https://openalex.org/keywords/euclidean-distance","display_name":"Euclidean distance","score":0.5898764729499817},{"id":"https://openalex.org/keywords/euclidean-distance-matrix","display_name":"Euclidean distance matrix","score":0.47808003425598145},{"id":"https://openalex.org/keywords/remainder","display_name":"Remainder","score":0.44410279393196106},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.34232738614082336},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.1321355402469635}],"concepts":[{"id":"https://openalex.org/C98371066","wikidata":"https://www.wikidata.org/wiki/Q230848","display_name":"Euclidean algorithm","level":2,"score":0.9368791580200195},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.8178026676177979},{"id":"https://openalex.org/C129782007","wikidata":"https://www.wikidata.org/wiki/Q162886","display_name":"Euclidean geometry","level":2,"score":0.6930805444717407},{"id":"https://openalex.org/C151635116","wikidata":"https://www.wikidata.org/wiki/Q193878","display_name":"Chinese remainder theorem","level":2,"score":0.63099604845047},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6220661401748657},{"id":"https://openalex.org/C23719512","wikidata":"https://www.wikidata.org/wiki/Q867345","display_name":"Euclidean domain","level":4,"score":0.6146183013916016},{"id":"https://openalex.org/C120174047","wikidata":"https://www.wikidata.org/wiki/Q847073","display_name":"Euclidean distance","level":2,"score":0.5898764729499817},{"id":"https://openalex.org/C171558263","wikidata":"https://www.wikidata.org/wiki/Q5406122","display_name":"Euclidean distance matrix","level":3,"score":0.47808003425598145},{"id":"https://openalex.org/C39613435","wikidata":"https://www.wikidata.org/wiki/Q846677","display_name":"Remainder","level":2,"score":0.44410279393196106},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.34232738614082336},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.1321355402469635},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.2478/v10037-012-0020-2","is_oa":true,"landing_page_url":"https://doi.org/10.2478/v10037-012-0020-2","pdf_url":"https://sciendo.com/pdf/10.2478/v10037-012-0020-2","source":{"id":"https://openalex.org/S179241345","display_name":"Formalized Mathematics","issn_l":"1426-2630","issn":["1426-2630","1898-9934"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320322","host_organization_name":"De Gruyter Open","host_organization_lineage":["https://openalex.org/P4310320322","https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter Open","De Gruyter"],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formalized Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:557c8403c66640a48e969c037e255ceb","is_oa":false,"landing_page_url":"https://doaj.org/article/557c8403c66640a48e969c037e255ceb","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Formalized Mathematics, Vol 20, Iss 2, Pp 175-179 (2012)","raw_type":"article"},{"id":"pmh:oai:repozytorium.uwb.edu.pl:11320/3640","is_oa":true,"landing_page_url":"http://hdl.handle.net/11320/3640","pdf_url":null,"source":{"id":"https://openalex.org/S4306400933","display_name":"The Repository of University in Bialystok (University in Bialystok)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210094198","host_organization_name":"University Clinical Hospital In Bialystok","host_organization_lineage":["https://openalex.org/I4210094198"],"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":null,"raw_type":"Article"},{"id":"pmh:oai:soar-ir.repo.nii.ac.jp:00018406","is_oa":true,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4306402150","display_name":"SOAR (Shinshu University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I137975476","host_organization_name":"Shinshu University","host_organization_lineage":["https://openalex.org/I137975476"],"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":"","raw_type":null}],"best_oa_location":{"id":"doi:10.2478/v10037-012-0020-2","is_oa":true,"landing_page_url":"https://doi.org/10.2478/v10037-012-0020-2","pdf_url":"https://sciendo.com/pdf/10.2478/v10037-012-0020-2","source":{"id":"https://openalex.org/S179241345","display_name":"Formalized Mathematics","issn_l":"1426-2630","issn":["1426-2630","1898-9934"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320322","host_organization_name":"De Gruyter Open","host_organization_lineage":["https://openalex.org/P4310320322","https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter Open","De Gruyter"],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formalized Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2083393265.pdf","grobid_xml":"https://content.openalex.org/works/W2083393265.grobid-xml"},"referenced_works_count":9,"referenced_works":["https://openalex.org/W192809544","https://openalex.org/W260340778","https://openalex.org/W1981663184","https://openalex.org/W2105559527","https://openalex.org/W2150348213","https://openalex.org/W2155453192","https://openalex.org/W2296232473","https://openalex.org/W2752061190","https://openalex.org/W4285719527"],"related_works":["https://openalex.org/W1537517674","https://openalex.org/W560234846","https://openalex.org/W1503447059","https://openalex.org/W4250443881","https://openalex.org/W4298160248","https://openalex.org/W2051374838","https://openalex.org/W2963523248","https://openalex.org/W2610477248","https://openalex.org/W2299429778","https://openalex.org/W2083393265"],"abstract_inverted_index":{"Summary":[0],"In":[1,43],"this":[2],"article":[3],"we":[4,45],"formalize":[5,46],"some":[6],"number":[7,72,90],"theoretical":[8],"algorithms,":[9],"Euclidean":[10,14,23,62],"Algorithm":[11,15,24],"and":[12],"Extended":[13,22,61],"[9].":[16],"Besides":[17],"the":[18,55,69,83,87],"a":[19,27,40,52,89],"gcd":[20,41],"b,":[21],"can":[25,50],"calculate":[26],"pair":[28],"of":[29,54,71,77,86],"two":[30],"integers":[31],"(x,":[32],"y)":[33],"that":[34,49],"holds":[35],"ax":[36],"+":[37],"by":[38,59,96],"=":[39],"b.":[42],"addition,":[44],"an":[47],"algorithm":[48],"compute":[51],"solution":[53],"Chinese":[56],"remainder":[57],"theorem":[58],"using":[60],"Algorithm.":[63],"Our":[64,75],"aim":[65],"is":[66,80],"to":[67],"support":[68],"implementation":[70],"theoretic":[73],"tools.":[74],"formalization":[76],"those":[78],"algorithms":[79],"based":[81],"on":[82],"source":[84],"code":[85],"NZMATH,":[88],"theory":[91],"oriented":[92],"calculation":[93],"system":[94],"developed":[95],"Tokyo":[97],"Metropolitan":[98],"University":[99],"[8].":[100]},"counts_by_year":[{"year":2019,"cited_by_count":1},{"year":2016,"cited_by_count":1}],"updated_date":"2026-07-02T09:51:11.867554","created_date":"2016-06-24T00:00:00"}
