{"id":"https://openalex.org/W2949119678","doi":"https://doi.org/10.3390/info10070232","title":"Idempotent Factorizations of Square-Free Integers","display_name":"Idempotent Factorizations of Square-Free Integers","publication_year":2019,"publication_date":"2019-07-06","ids":{"openalex":"https://openalex.org/W2949119678","doi":"https://doi.org/10.3390/info10070232","mag":"2949119678"},"language":"en","primary_location":{"id":"doi:10.3390/info10070232","is_oa":true,"landing_page_url":"https://doi.org/10.3390/info10070232","pdf_url":"https://www.mdpi.com/2078-2489/10/7/232/pdf?version=1562405229","source":{"id":"https://openalex.org/S4210219776","display_name":"Information","issn_l":"2078-2489","issn":["2078-2489"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Information","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2078-2489/10/7/232/pdf?version=1562405229","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5063844820","display_name":"Barry Fagin","orcid":"https://orcid.org/0000-0001-5940-2598"},"institutions":[{"id":"https://openalex.org/I430641","display_name":"United States Air Force Academy","ror":"https://ror.org/0055d0g64","country_code":"US","type":"government","lineage":["https://openalex.org/I1330347796","https://openalex.org/I4210089612","https://openalex.org/I4210102105","https://openalex.org/I430641"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Barry Fagin","raw_affiliation_strings":["Department of Computer Science, US Air Force Academy, Colorado Springs, CO 80840, USA"],"raw_orcid":"https://orcid.org/0000-0001-5940-2598","affiliations":[{"raw_affiliation_string":"Department of Computer Science, US Air Force Academy, Colorado Springs, CO 80840, USA","institution_ids":["https://openalex.org/I430641"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5063844820"],"corresponding_institution_ids":["https://openalex.org/I430641"],"apc_list":{"value":1600,"currency":"CHF","value_usd":1782},"apc_paid":{"value":1600,"currency":"CHF","value_usd":1782},"fwci":0.1828,"has_fulltext":true,"cited_by_count":3,"citation_normalized_percentile":{"value":0.49077491,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"10","issue":"7","first_page":"232","last_page":"232"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11797","display_name":"graph theory and CDMA systems","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11797","display_name":"graph theory and CDMA systems","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.9994000196456909,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11166","display_name":"Analytic Number Theory Research","score":0.9965999722480774,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/idempotence","display_name":"Idempotence","score":0.8047110438346863},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7769107818603516},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5873318314552307},{"id":"https://openalex.org/keywords/idempotent-matrix","display_name":"Idempotent matrix","score":0.5620971322059631},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.5382267236709595},{"id":"https://openalex.org/keywords/prime","display_name":"Prime (order theory)","score":0.5161184072494507},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.4963141083717346},{"id":"https://openalex.org/keywords/square-free-integer","display_name":"Square-free integer","score":0.48404043912887573},{"id":"https://openalex.org/keywords/factorization","display_name":"Factorization","score":0.44356557726860046},{"id":"https://openalex.org/keywords/ring","display_name":"Ring (chemistry)","score":0.4129924774169922},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.1283816695213318},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.1212630569934845}],"concepts":[{"id":"https://openalex.org/C172252984","wikidata":"https://www.wikidata.org/wiki/Q368988","display_name":"Idempotence","level":2,"score":0.8047110438346863},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7769107818603516},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5873318314552307},{"id":"https://openalex.org/C33375987","wikidata":"https://www.wikidata.org/wiki/Q3751720","display_name":"Idempotent matrix","level":3,"score":0.5620971322059631},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5382267236709595},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.5161184072494507},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.4963141083717346},{"id":"https://openalex.org/C188139237","wikidata":"https://www.wikidata.org/wiki/Q50706","display_name":"Square-free integer","level":2,"score":0.48404043912887573},{"id":"https://openalex.org/C187834632","wikidata":"https://www.wikidata.org/wiki/Q188804","display_name":"Factorization","level":2,"score":0.44356557726860046},{"id":"https://openalex.org/C2780378348","wikidata":"https://www.wikidata.org/wiki/Q25351438","display_name":"Ring (chemistry)","level":2,"score":0.4129924774169922},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.1283816695213318},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.1212630569934845},{"id":"https://openalex.org/C178790620","wikidata":"https://www.wikidata.org/wiki/Q11351","display_name":"Organic chemistry","level":1,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/info10070232","is_oa":true,"landing_page_url":"https://doi.org/10.3390/info10070232","pdf_url":"https://www.mdpi.com/2078-2489/10/7/232/pdf?version=1562405229","source":{"id":"https://openalex.org/S4210219776","display_name":"Information","issn_l":"2078-2489","issn":["2078-2489"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Information","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:7ab4f52f44a249f9999ada70b93c472d","is_oa":true,"landing_page_url":"https://doaj.org/article/7ab4f52f44a249f9999ada70b93c472d","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":"Information, Vol 10, Iss 7, p 232 (2019)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2078-2489/10/7/232/","is_oa":true,"landing_page_url":"http://dx.doi.org/10.3390/info10070232","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Information","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/info10070232","is_oa":true,"landing_page_url":"https://doi.org/10.3390/info10070232","pdf_url":"https://www.mdpi.com/2078-2489/10/7/232/pdf?version=1562405229","source":{"id":"https://openalex.org/S4210219776","display_name":"Information","issn_l":"2078-2489","issn":["2078-2489"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Information","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320337633","display_name":"U.S. Air Force Academy","ror":"https://ror.org/0055d0g64"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2949119678.pdf","grobid_xml":"https://content.openalex.org/works/W2949119678.grobid-xml"},"referenced_works_count":8,"referenced_works":["https://openalex.org/W175229225","https://openalex.org/W2013805494","https://openalex.org/W2047316746","https://openalex.org/W2126209209","https://openalex.org/W2889140723","https://openalex.org/W2915617585","https://openalex.org/W4232836212","https://openalex.org/W6672463992"],"related_works":["https://openalex.org/W4212917286","https://openalex.org/W2379578187","https://openalex.org/W1979565872","https://openalex.org/W2357199145","https://openalex.org/W2393613245","https://openalex.org/W2389832915","https://openalex.org/W2381440097","https://openalex.org/W4303633560","https://openalex.org/W2350953973","https://openalex.org/W4287979965"],"abstract_inverted_index":{"We":[0,154,160,178],"explore":[1],"the":[2,41,56,99,107,110,142,163],"class":[3],"of":[4,70,144,165],"positive":[5],"integers":[6],"n":[7,12,22,71,105,170],"that":[8,19,93,128],"admit":[9],"idempotent":[10,73,130,167],"factorizations":[11,46,74,84,131,175],"=":[13],"p":[14,26,48,77,88,112],"\u00af":[15,17,27,33,49,52,78,81,89,92,113,116],"q":[16,32,51,80,91,115],"such":[18],"\u03bb":[20,39],"(":[21,25,31],")":[23,30,36],"\u2223":[24],"\u2212":[28,34],"1":[29,35],",":[37],"where":[38],"is":[40],"Carmichael":[42],"lambda":[43],"function.":[44],"Idempotent":[45,83],"with":[47,72,104],"and":[50,90,114,121,151,184],"prime":[53],"have":[54,117],"received":[55],"most":[57],"attention":[58],"due":[59],"to":[60,181],"their":[61,135,189],"cryptographic":[62,119],"advantages,":[63],"but":[64],"there":[65],"are":[66,85,176],"an":[67],"infinite":[68],"number":[69,152],"containing":[75],"composite":[76],"and/or":[79],".":[82],"exactly":[86],"those":[87,169],"generate":[94],"correctly":[95],"functioning":[96],"keys":[97],"in":[98,127,134,148],"Rivest\u2013Shamir\u2013Adleman":[100],"(RSA)":[101],"2-prime":[102],"protocol":[103],"as":[106,138],"modulus.":[108],"While":[109],"resulting":[111],"no":[118],"utility":[120],"therefore":[122],"should":[123],"never":[124],"be":[125],"employed":[126],"capacity,":[129],"warrant":[132],"study":[133],"own":[136],"right":[137],"they":[139],"live":[140],"at":[141],"intersection":[143],"multiple":[145],"hard":[146],"problems":[147],"computer":[149],"science":[150],"theory.":[153],"present":[155,185],"some":[156],"analytical":[157],"results":[158,187],"here.":[159],"also":[161],"demonstrate":[162],"existence":[164],"maximally":[166],"integers,":[168],"for":[171],"which":[172],"all":[173],"bipartite":[174],"idempotent.":[177],"show":[179],"how":[180],"construct":[182],"them,":[183],"preliminary":[186],"on":[188],"distribution.":[190]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":1}],"updated_date":"2026-07-23T08:03:31.855105","created_date":"2025-10-10T00:00:00"}
