{"id":"https://openalex.org/W2784118590","doi":"https://doi.org/10.1090/mcom/3349","title":"Dirichlet\u2019s proof of the three-square theorem: An algorithmic perspective","display_name":"Dirichlet\u2019s proof of the three-square theorem: An algorithmic perspective","publication_year":2018,"publication_date":"2018-01-10","ids":{"openalex":"https://openalex.org/W2784118590","doi":"https://doi.org/10.1090/mcom/3349","mag":"2784118590"},"language":"en","primary_location":{"id":"doi:10.1090/mcom/3349","is_oa":true,"landing_page_url":"https://doi.org/10.1090/mcom/3349","pdf_url":"https://www.ams.org/mcom/2019-88-316/S0025-5718-2018-03349-X/S0025-5718-2018-03349-X.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/2019-88-316/S0025-5718-2018-03349-X/S0025-5718-2018-03349-X.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5089571181","display_name":"Paul Pollack","orcid":"https://orcid.org/0000-0002-4327-5560"},"institutions":[{"id":"https://openalex.org/I165733156","display_name":"University of Georgia","ror":"https://ror.org/00te3t702","country_code":"US","type":"education","lineage":["https://openalex.org/I165733156"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Paul Pollack","raw_affiliation_strings":["Department of Mathematics, University of Georgia, Athens, Georgia 30602"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Georgia, Athens, Georgia 30602","institution_ids":["https://openalex.org/I165733156"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5016212896","display_name":"Peter Schorn","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Peter Schorn","raw_affiliation_strings":["Culmannstrasse 77, CH-8006 Zurich, Switzerland"],"affiliations":[{"raw_affiliation_string":"Culmannstrasse 77, CH-8006 Zurich, Switzerland","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5089571181"],"corresponding_institution_ids":["https://openalex.org/I165733156"],"apc_list":null,"apc_paid":null,"fwci":0.965,"has_fulltext":true,"cited_by_count":7,"citation_normalized_percentile":{"value":0.69396742,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"88","issue":"316","first_page":"1007","last_page":"1019"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11166","display_name":"Analytic Number Theory Research","score":0.9998999834060669,"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":0.9998999834060669,"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9991000294685364,"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"}},{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.9987999796867371,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.9266977310180664},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6295408606529236},{"id":"https://openalex.org/keywords/analytic-number-theory","display_name":"Analytic number theory","score":0.6162996292114258},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.613505482673645},{"id":"https://openalex.org/keywords/dirichlet-distribution","display_name":"Dirichlet distribution","score":0.6000100374221802},{"id":"https://openalex.org/keywords/square","display_name":"Square (algebra)","score":0.49935412406921387},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4592340886592865},{"id":"https://openalex.org/keywords/quadratic-integer","display_name":"Quadratic integer","score":0.4402504563331604},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.09700566530227661}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.9266977310180664},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6295408606529236},{"id":"https://openalex.org/C149381693","wikidata":"https://www.wikidata.org/wiki/Q10843274","display_name":"Analytic number theory","level":2,"score":0.6162996292114258},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.613505482673645},{"id":"https://openalex.org/C169214877","wikidata":"https://www.wikidata.org/wiki/Q981016","display_name":"Dirichlet distribution","level":3,"score":0.6000100374221802},{"id":"https://openalex.org/C135692309","wikidata":"https://www.wikidata.org/wiki/Q111124","display_name":"Square (algebra)","level":2,"score":0.49935412406921387},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4592340886592865},{"id":"https://openalex.org/C19875773","wikidata":"https://www.wikidata.org/wiki/Q803531","display_name":"Quadratic integer","level":3,"score":0.4402504563331604},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.09700566530227661},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.0},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1090/mcom/3349","is_oa":true,"landing_page_url":"https://doi.org/10.1090/mcom/3349","pdf_url":"https://www.ams.org/mcom/2019-88-316/S0025-5718-2018-03349-X/S0025-5718-2018-03349-X.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"}],"best_oa_location":{"id":"doi:10.1090/mcom/3349","is_oa":true,"landing_page_url":"https://doi.org/10.1090/mcom/3349","pdf_url":"https://www.ams.org/mcom/2019-88-316/S0025-5718-2018-03349-X/S0025-5718-2018-03349-X.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":[{"id":"https://openalex.org/G4665435243","display_name":null,"funder_award_id":"DMS-1402268","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2784118590.pdf","grobid_xml":"https://content.openalex.org/works/W2784118590.grobid-xml"},"referenced_works_count":30,"referenced_works":["https://openalex.org/W1507575253","https://openalex.org/W1563414093","https://openalex.org/W1566908256","https://openalex.org/W1754110195","https://openalex.org/W2002059900","https://openalex.org/W2025525128","https://openalex.org/W2026360568","https://openalex.org/W2037038460","https://openalex.org/W2037363226","https://openalex.org/W2043545422","https://openalex.org/W2050369625","https://openalex.org/W2071743058","https://openalex.org/W2077098038","https://openalex.org/W2087957223","https://openalex.org/W2127973214","https://openalex.org/W2131286165","https://openalex.org/W2156734628","https://openalex.org/W2318257401","https://openalex.org/W2318326253","https://openalex.org/W2334411759","https://openalex.org/W2963464756","https://openalex.org/W2963623033","https://openalex.org/W3013275592","https://openalex.org/W3101181971","https://openalex.org/W3142624779","https://openalex.org/W6600234944","https://openalex.org/W6600387335","https://openalex.org/W6631002067","https://openalex.org/W6806482286","https://openalex.org/W6824070958"],"related_works":["https://openalex.org/W4246602673","https://openalex.org/W1514073005","https://openalex.org/W2012867998","https://openalex.org/W4255421128","https://openalex.org/W1970308065","https://openalex.org/W4286908302","https://openalex.org/W4320461947","https://openalex.org/W3185358151","https://openalex.org/W2950914967","https://openalex.org/W2346447845"],"abstract_inverted_index":{"The":[0],"Gauss\u00e2\u0080\u0093Legendre":[1],"three-square":[2],"theorem":[3],"asserts":[4],"that":[5],"the":[6,23,63,77,97,128],"positive":[7],"integers":[8,29],"$n$":[9],"expressible":[10],"as":[11],"a":[12,35,70],"sum":[13],"of":[14,22,39,80,100,111,134],"three":[15],"integer":[16],"squares":[17],"are":[18],"precisely":[19],"those":[20],"not":[21],"form":[24],"$4^k(8m+7)$":[25],"for":[26,73,96,108],"any":[27],"nonnegative":[28],"$k,m$.":[30],"In":[31],"1850,":[32],"Dirichlet":[33],"gave":[34],"beautifully":[36],"simple":[37],"proof":[38,56],"this":[40,109],"result":[41],"using":[42],"only":[43],"basic":[44],"facts":[45],"about":[46],"ternary":[47],"quadratic":[48],"forms.":[49],"We":[50],"explain":[51],"how":[52],"to":[53],"turn":[54],"Dirichlet\u00e2\u0080\u0099s":[55],"into":[57],"an":[58],"algorithm;":[59],"if":[60],"one":[61],"assumes":[62],"Extended":[64],"Riemann":[65],"Hypothesis":[66],"(ERH),":[67],"there":[68],"is":[69,83],"random":[71,106],"algorithm":[72,107],"expressing":[74],"$n=x^2+y^2+z^2$,":[75],"where":[76],"expected":[78],"number":[79],"bit":[81,98],"operations":[82],"$O((\\lg":[84],"n)^2":[85],"(\\lg":[86],"\\lg":[87],"n)^{-1}":[88],"\\cdot":[89],"M(\\lg":[90],"n))$.":[91],"Here,":[92],"$M(r)$":[93],"stands":[94],"in":[95,120],"complexity":[99,113],"multiplying":[101],"two":[102],"$r$-bit":[103],"integers.":[104],"A":[105],"problem":[110],"similar":[112],"was":[114],"proposed":[115],"by":[116],"Rabin":[117],"and":[118,130],"Shallit":[119],"1986;":[121],"however,":[122],"their":[123],"analysis":[124],"depended":[125],"on":[126,131],"both":[127],"ERH":[129],"certain":[132],"conjectures":[133],"Hardy\u00e2\u0080\u0093Littlewood":[135],"type.":[136]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}