{"id":"https://openalex.org/W1988839728","doi":"https://doi.org/10.1090/s0025-5718-96-00725-9","title":"Density computations for real quadratic units","display_name":"Density computations for real quadratic units","publication_year":1996,"publication_date":"1996-01-01","ids":{"openalex":"https://openalex.org/W1988839728","doi":"https://doi.org/10.1090/s0025-5718-96-00725-9","mag":"1988839728"},"language":"lv","primary_location":{"id":"doi:10.1090/s0025-5718-96-00725-9","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-96-00725-9","pdf_url":"https://www.ams.org/mcom/1996-65-215/S0025-5718-96-00725-9/S0025-5718-96-00725-9.pdf","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":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://www.ams.org/mcom/1996-65-215/S0025-5718-96-00725-9/S0025-5718-96-00725-9.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5109232255","display_name":"Wieb Bosma","orcid":null},"institutions":[{"id":"https://openalex.org/I129604602","display_name":"The University of Sydney","ror":"https://ror.org/0384j8v12","country_code":"AU","type":"education","lineage":["https://openalex.org/I129604602"]}],"countries":["AU"],"is_corresponding":true,"raw_author_name":"Wieb Bosma","raw_affiliation_strings":["School of Mathematics and Statistics, University of Sydney, Sydney NSW 2006, Australia","School of Mathematics and Statistics, University of Sydney, Sydney, NSW 2006, Australia ("],"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics, University of Sydney, Sydney NSW 2006, Australia","institution_ids":["https://openalex.org/I129604602"]},{"raw_affiliation_string":"School of Mathematics and Statistics, University of Sydney, Sydney, NSW 2006, Australia (","institution_ids":["https://openalex.org/I129604602"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5008644708","display_name":"Peter Stevenhagen","orcid":null},"institutions":[{"id":"https://openalex.org/I1341640284","display_name":"Centrum Wiskunde & Informatica","ror":"https://ror.org/00x7ekv49","country_code":"NL","type":"facility","lineage":["https://openalex.org/I1341640284","https://openalex.org/I2800991832","https://openalex.org/I4405262988"]},{"id":"https://openalex.org/I887064364","display_name":"University of Amsterdam","ror":"https://ror.org/04dkp9463","country_code":"NL","type":"education","lineage":["https://openalex.org/I887064364"]}],"countries":["NL"],"is_corresponding":false,"raw_author_name":"Peter Stevenhagen","raw_affiliation_strings":["Faculteit Wiskunde en Informatica, Universiteit van Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands","Faculteit Wiskunde en Informatica, Universiteit van Amsterdam, Plantage Muidergracht 24, 1018 TV, Amsterdam, The Netherlands"],"affiliations":[{"raw_affiliation_string":"Faculteit Wiskunde en Informatica, Universiteit van Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands","institution_ids":["https://openalex.org/I1341640284","https://openalex.org/I887064364"]},{"raw_affiliation_string":"Faculteit Wiskunde en Informatica, Universiteit van Amsterdam, Plantage Muidergracht 24, 1018 TV, Amsterdam, The Netherlands","institution_ids":["https://openalex.org/I1341640284","https://openalex.org/I887064364"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5109232255"],"corresponding_institution_ids":["https://openalex.org/I129604602"],"apc_list":null,"apc_paid":null,"fwci":0.9789,"has_fulltext":true,"cited_by_count":6,"citation_normalized_percentile":{"value":0.71613663,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":95},"biblio":{"volume":"65","issue":"215","first_page":"1327","last_page":"1337"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11476","display_name":"Graph theory and applications","score":0.9998000264167786,"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/T11476","display_name":"Graph theory and applications","score":0.9998000264167786,"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"}},{"id":"https://openalex.org/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.9994999766349792,"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"}},{"id":"https://openalex.org/T10849","display_name":"Finite Group Theory Research","score":0.9977999925613403,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/algorithm","display_name":"Algorithm","score":0.6757374405860901},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.39698076248168945},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.3626803755760193}],"concepts":[{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.6757374405860901},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.39698076248168945},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.3626803755760193}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1090/s0025-5718-96-00725-9","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-96-00725-9","pdf_url":"https://www.ams.org/mcom/1996-65-215/S0025-5718-96-00725-9/S0025-5718-96-00725-9.pdf","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:dare.uva.nl:openaire_cris_publications/ae07762f-a082-4af0-94ae-362caef69e2d","is_oa":false,"landing_page_url":"https://dare.uva.nl/personal/pure/en/publications/density-computations-for-real-quadratic-units(ae07762f-a082-4af0-94ae-362caef69e2d).html","pdf_url":null,"source":{"id":"https://openalex.org/S4306400088","display_name":"UvA-DARE (University of Amsterdam)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I887064364","host_organization_name":"University of Amsterdam","host_organization_lineage":["https://openalex.org/I887064364"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Bosma, W & Stevenhagen, P 1996, 'Density computations for real quadratic units', Mathematics of Computation, vol. 65, no. 215, pp. 1327-1337. https://doi.org/10.1090/S0025-5718-96-00725-9","raw_type":"info:eu-repo/semantics/publishedVersion"},{"id":"pmh:oai:dare.uva.nl:publications/ae07762f-a082-4af0-94ae-362caef69e2d","is_oa":false,"landing_page_url":"https://handle.uba.uva.nl/personal/pure/en/publications/density-computations-for-real-quadratic-units(ae07762f-a082-4af0-94ae-362caef69e2d).html","pdf_url":null,"source":{"id":"https://openalex.org/S4306400088","display_name":"UvA-DARE (University of Amsterdam)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I887064364","host_organization_name":"University of Amsterdam","host_organization_lineage":["https://openalex.org/I887064364"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Bosma, W & Stevenhagen, P 1996, 'Density computations for real quadratic units', Mathematics of Computation, vol. 65, no. 215, pp. 1327-1337. https://doi.org/10.1090/S0025-5718-96-00725-9","raw_type":"info:eu-repo/semantics/publishedVersion"}],"best_oa_location":{"id":"doi:10.1090/s0025-5718-96-00725-9","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-96-00725-9","pdf_url":"https://www.ams.org/mcom/1996-65-215/S0025-5718-96-00725-9/S0025-5718-96-00725-9.pdf","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"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1988839728.pdf","grobid_xml":"https://content.openalex.org/works/W1988839728.grobid-xml"},"referenced_works_count":11,"referenced_works":["https://openalex.org/W57434570","https://openalex.org/W1479719134","https://openalex.org/W1496801598","https://openalex.org/W1983003491","https://openalex.org/W2025525128","https://openalex.org/W2077098038","https://openalex.org/W3113374636","https://openalex.org/W4246973793","https://openalex.org/W4249765387","https://openalex.org/W4251287813","https://openalex.org/W4285719527"],"related_works":["https://openalex.org/W2748952813","https://openalex.org/W2051487156","https://openalex.org/W2073681303","https://openalex.org/W2390279801","https://openalex.org/W2358668433","https://openalex.org/W2376932109","https://openalex.org/W2001405890","https://openalex.org/W2382290278","https://openalex.org/W2350741829","https://openalex.org/W2530322880"],"abstract_inverted_index":{"In":[0,193],"order":[1],"to":[2,311],"study":[3],"the":[4,7,26,80,83,132,152,155,180,184,190,302],"density":[5,181],"of":[6,9,82,90,134,151,157,301,320],"set":[8],"positive":[10],"integers":[11],"<inline-formula":[12,30,111,135,158,198,322],"content-type=\"math/mathml\">":[13,31,112,136,159,199,323],"<mml:math":[14,32,113,137,160,200,324],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[15,33,114,138,161,201,325],"alttext=\"d\">":[16,139,162],"<mml:semantics>":[17,43,117,140,163,230,328],"<mml:mi>d</mml:mi>":[18,55,123,141,164],"<mml:annotation":[19,69,125,142,165,285,335],"encoding=\"application/x-tex\">d</mml:annotation>":[20,143,166],"</mml:semantics>":[21,71,128,144,167,298,337],"</mml:math>":[22,72,129,145,168,299,338],"</inline-formula>":[23,73,130,146,169,300,339],"for":[24,154,179,183],"which":[25],"negative":[27,41,185,224],"Pell":[28,186],"equation":[29,187],"alttext=\"x":[34],"squared":[35,39],"minus":[36,206,221],"d":[37,215,217],"y":[38],"equals":[40,204],"1\">":[42,327],"<mml:mrow>":[44,118,231,329],"<mml:msup>":[45,56,268],"<mml:mi>x</mml:mi>":[46],"<mml:mrow":[47,58,242,251,270],"class=\"MJX-TeXAtom-ORD\">":[48,59,243,252,271],"<mml:mn>2</mml:mn>":[49,60,269],"</mml:mrow>":[50,61,68,124,259,260,276,284,334],"</mml:msup>":[51,62,277],"<mml:mo>":[52,64,120,235,239,245,265,272,280,330],"\u2212":[53,65,236,266,273,331],"</mml:mo>":[54,66,122,237,241,247,267,274,282,332],"<mml:mi>y</mml:mi>":[57],"<mml:mo>=</mml:mo>":[63,233],"<mml:mn>1</mml:mn>":[67,234,248,264,333],"encoding=\"application/x-tex\">x^{2}-dy^{2}=-1</mml:annotation>":[70],"is":[74,104,107,147],"solvable":[75],"in":[76,86,110],"integers,":[77],"we":[78,170],"compute":[79],"norm":[81,321],"fundamental":[84,318],"unit":[85,319],"certain":[87],"well-chosen":[88],"families":[89],"real":[91,303],"quadratic":[92,304],"orders.":[93],"A":[94],"fast":[95],"algorithm":[96],"that":[97],"computes":[98],"2-class":[99],"groups":[100],"rather":[101],"than":[102],"units":[103],"used.":[105],"It":[106],"random":[108],"polynomial-time":[109],"alttext=\"log":[115],"d\">":[116],"<mml:mi>log</mml:mi>":[119],"\u2061":[121],"encoding=\"application/x-tex\">\\log":[126],"d</mml:annotation>":[127],"as":[131],"factorization":[133],"a":[148,317],"natural":[149],"part":[150],"input":[153],"values":[156],"encounter.":[171],"The":[172],"data":[173],"obtained":[174],"provide":[175],"convincing":[176],"numerical":[177],"evidence":[178],"heuristics":[182],"proposed":[188],"by":[189],"second":[191],"author.":[192],"particular,":[194],"an":[195],"irrational":[196],"proportion":[197],"alttext=\"upper":[202],"P":[203],"1":[205,211,220,288],"product":[207],"Underscript":[208],"j":[209,225],"greater-than-or-equal-to":[210],"normal":[212,214,216],"o":[213],"Endscripts":[218],"left-parenthesis":[219],"2":[222],"Superscript":[223],"Baseline":[226],"right-parenthesis":[227],"almost-equals":[228],".58\">":[229],"<mml:mi>P</mml:mi>":[232],"<mml:munder>":[238],"\u220f":[240],"<mml:mi>j</mml:mi>":[244,275],"\u2265":[246],"<mml:mspace":[249],"width=\"thickmathspace\"/>":[250],"<mml:mi":[253,255,257],"mathvariant=\"normal\">o</mml:mi>":[254],"mathvariant=\"normal\">d</mml:mi>":[256,258],"</mml:munder>":[261],"<mml:mo":[262,278],"stretchy=\"false\">(</mml:mo>":[263],"stretchy=\"false\">)</mml:mo>":[279],"\u2248":[281],"<mml:mn>.58</mml:mn>":[283],"encoding=\"application/x-tex\">P":[286],"=":[287],"-":[289],"\\prod":[290],"_{j\\ge":[291],"1\\;":[292],"\\mathrm":[293],"{odd}}":[294],"(1-2^{-j})":[295],"\\approx":[296],".58</mml:annotation>":[297],"fields":[305],"without":[306],"discriminantal":[307],"prime":[308],"divisors":[309],"congruent":[310],"3":[312],"mod":[313],"4":[314],"should":[315],"have":[316],"alttext=\"negative":[326],"encoding=\"application/x-tex\">-1</mml:annotation>":[336],".":[340]},"counts_by_year":[{"year":2021,"cited_by_count":2},{"year":2015,"cited_by_count":1}],"updated_date":"2026-04-21T06:05:55.347736","created_date":"2025-10-10T00:00:00"}