{"id":"https://openalex.org/W2040928009","doi":"https://doi.org/10.1090/s0025-5718-2010-02381-6","title":"On the largest prime factor of $x^{2}-1$","display_name":"On the largest prime factor of $x^{2}-1$","publication_year":2010,"publication_date":"2010-07-20","ids":{"openalex":"https://openalex.org/W2040928009","doi":"https://doi.org/10.1090/s0025-5718-2010-02381-6","mag":"2040928009"},"language":"en","primary_location":{"id":"doi:10.1090/s0025-5718-2010-02381-6","is_oa":false,"landing_page_url":"https://doi.org/10.1090/s0025-5718-2010-02381-6","pdf_url":null,"source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"type":"preprint","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1005.1533","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5007752336","display_name":"Florian Luca","orcid":"https://orcid.org/0000-0003-1321-4422"},"institutions":[{"id":"https://openalex.org/I8961855","display_name":"Universidad Nacional Aut\u00f3noma de M\u00e9xico","ror":"https://ror.org/01tmp8f25","country_code":"MX","type":"education","lineage":["https://openalex.org/I8961855"]}],"countries":["MX"],"is_corresponding":true,"raw_author_name":"Florian Luca","raw_affiliation_strings":["Instituto de Matematicas, Universidad Nacional Autonoma de Mexico, C.P. 58089, Morelia, Michoacan, Mexico","Instituto de Matem\u00e1ticas, Universidad Nacional Aut\u00f3noma de Mexico, C.P. 58089, Morelia, Michoac\u00e1n, Mexico"],"affiliations":[{"raw_affiliation_string":"Instituto de Matematicas, Universidad Nacional Autonoma de Mexico, C.P. 58089, Morelia, Michoacan, Mexico","institution_ids":["https://openalex.org/I8961855"]},{"raw_affiliation_string":"Instituto de Matem\u00e1ticas, Universidad Nacional Aut\u00f3noma de Mexico, C.P. 58089, Morelia, Michoac\u00e1n, Mexico","institution_ids":["https://openalex.org/I8961855"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5081214859","display_name":"Filip Najman","orcid":"https://orcid.org/0000-0002-0994-0846"},"institutions":[{"id":"https://openalex.org/I181343428","display_name":"University of Zagreb","ror":"https://ror.org/00mv6sv71","country_code":"HR","type":"education","lineage":["https://openalex.org/I181343428"]}],"countries":["HR"],"is_corresponding":false,"raw_author_name":"Filip Najman","raw_affiliation_strings":["Department of Mathematics, University of Zagreb, Bijeni\u00c4\u008dka Cesta 30, 10000 Zagreb, Croatia","Department of Mathematics, University of Zagreb, Bijenicka Cesta 30, 10000 Zagreb, Croatia"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Zagreb, Bijeni\u00c4\u008dka Cesta 30, 10000 Zagreb, Croatia","institution_ids":["https://openalex.org/I181343428"]},{"raw_affiliation_string":"Department of Mathematics, University of Zagreb, Bijenicka Cesta 30, 10000 Zagreb, Croatia","institution_ids":["https://openalex.org/I181343428"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5007752336"],"corresponding_institution_ids":["https://openalex.org/I8961855"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.15198574,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"80","issue":"273","first_page":"429","last_page":"435"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11166","display_name":"Analytic Number Theory Research","score":0.977400004863739,"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":0.977400004863739,"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/T11703","display_name":"Rings, Modules, and Algebras","score":0.9664999842643738,"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/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.9549000263214111,"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/prime-factor","display_name":"Prime factor","score":0.7559581995010376},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.7290859818458557},{"id":"https://openalex.org/keywords/prime","display_name":"Prime (order theory)","score":0.7008450031280518},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5786644220352173},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5384871959686279},{"id":"https://openalex.org/keywords/factor","display_name":"Factor (programming language)","score":0.440470427274704},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.39524519443511963},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.16189688444137573}],"concepts":[{"id":"https://openalex.org/C30860621","wikidata":"https://www.wikidata.org/wiki/Q49008","display_name":"Prime factor","level":3,"score":0.7559581995010376},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.7290859818458557},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.7008450031280518},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5786644220352173},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5384871959686279},{"id":"https://openalex.org/C2781039887","wikidata":"https://www.wikidata.org/wiki/Q1391724","display_name":"Factor (programming language)","level":2,"score":0.440470427274704},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.39524519443511963},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.16189688444137573},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1090/s0025-5718-2010-02381-6","is_oa":false,"landing_page_url":"https://doi.org/10.1090/s0025-5718-2010-02381-6","pdf_url":null,"source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1005.1533","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1005.1533","pdf_url":"https://arxiv.org/pdf/1005.1533","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"text"},{"id":"mag:2040928009","is_oa":true,"landing_page_url":"https://arxiv.org/pdf/1005.1533.pdf","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"arXiv (Cornell University)","raw_type":null},{"id":"doi:10.48550/arxiv.1005.1533","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1005.1533","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article-journal"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1005.1533","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1005.1533","pdf_url":"https://arxiv.org/pdf/1005.1533","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"text"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":2,"referenced_works":["https://openalex.org/W1708838831","https://openalex.org/W2051399739"],"related_works":["https://openalex.org/W2745843585","https://openalex.org/W2248539817","https://openalex.org/W2075140667","https://openalex.org/W2267407109","https://openalex.org/W2506905834","https://openalex.org/W1588000126","https://openalex.org/W2963670015","https://openalex.org/W2107800009","https://openalex.org/W2292792029","https://openalex.org/W2789708245","https://openalex.org/W2379608170","https://openalex.org/W2047138216","https://openalex.org/W2315667845","https://openalex.org/W3137852362","https://openalex.org/W2737617634","https://openalex.org/W2005901560","https://openalex.org/W2587693552","https://openalex.org/W2907773479","https://openalex.org/W2835095737","https://openalex.org/W2949190141"],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"we":[3,31],"find":[4,33],"all":[5,41],"integers":[6],"$x$":[7,38],"such":[8,39],"that":[9,40],"$x^{2}-1$":[10],"has":[11],"only":[12],"prime":[13,42],"factors":[14,43],"smaller":[15],"than":[16,53],"$100$.":[17],"This":[18],"gives":[19],"some":[20],"interesting":[21],"numerical":[22],"corollaries.":[23],"For":[24],"example,":[25],"for":[26],"any":[27],"positive":[28,36],"integer":[29,37],"$n$":[30],"can":[32],"the":[34],"largest":[35],"of":[44,46],"each":[45],"$x,":[47],"x+1,\\ldots":[48],",":[49],"x+n$":[50],"are":[51],"less":[52],"100.":[54]},"counts_by_year":[],"updated_date":"2026-03-10T16:38:18.471706","created_date":"2025-10-10T00:00:00"}