{"id":"https://openalex.org/W1972837770","doi":"https://doi.org/10.1090/s0025-5718-00-01290-4","title":"Verifying the Goldbach conjecture up to 4\u22c510\u00b9\u2074","display_name":"Verifying the Goldbach conjecture up to 4\u22c510\u00b9\u2074","publication_year":2000,"publication_date":"2000-07-18","ids":{"openalex":"https://openalex.org/W1972837770","doi":"https://doi.org/10.1090/s0025-5718-00-01290-4","mag":"1972837770"},"language":"en","primary_location":{"id":"doi:10.1090/s0025-5718-00-01290-4","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-00-01290-4","pdf_url":"https://www.ams.org/mcom/2001-70-236/S0025-5718-00-01290-4/S0025-5718-00-01290-4.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/2001-70-236/S0025-5718-00-01290-4/S0025-5718-00-01290-4.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5055287940","display_name":"J. Richstein","orcid":"https://orcid.org/0009-0006-2646-6249"},"institutions":[{"id":"https://openalex.org/I200763008","display_name":"Justus-Liebig-Universit\u00e4t Gie\u00dfen","ror":"https://ror.org/033eqas34","country_code":"DE","type":"education","lineage":["https://openalex.org/I200763008"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"J\u00f6rg Richstein","raw_affiliation_strings":["Institut f\u00c3\u00bcr Informatik, Fachbereich Mathematik, Justus-Liebig-Universit\u00c3\u00a4t, Gies- sen, Germany","Institut f\u00fcr Informatik, Fachbereich Mathematik, Justus-Liebig-Universit\u00e4t, Gies- sen, Germany"],"affiliations":[{"raw_affiliation_string":"Institut f\u00c3\u00bcr Informatik, Fachbereich Mathematik, Justus-Liebig-Universit\u00c3\u00a4t, Gies- sen, Germany","institution_ids":["https://openalex.org/I200763008"]},{"raw_affiliation_string":"Institut f\u00fcr Informatik, Fachbereich Mathematik, Justus-Liebig-Universit\u00e4t, Gies- sen, Germany","institution_ids":["https://openalex.org/I200763008"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5055287940"],"corresponding_institution_ids":["https://openalex.org/I200763008"],"apc_list":null,"apc_paid":null,"fwci":2.6259,"has_fulltext":true,"cited_by_count":64,"citation_normalized_percentile":{"value":0.87494892,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"70","issue":"236","first_page":"1745","last_page":"1749"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11166","display_name":"Analytic Number Theory Research","score":0.9995999932289124,"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":0.9995999932289124,"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/T10849","display_name":"Finite Group Theory Research","score":0.9904000163078308,"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9861999750137329,"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":0.6848286986351013},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.47103315591812134},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.36652374267578125}],"concepts":[{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.6848286986351013},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.47103315591812134},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.36652374267578125}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1090/s0025-5718-00-01290-4","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-00-01290-4","pdf_url":"https://www.ams.org/mcom/2001-70-236/S0025-5718-00-01290-4/S0025-5718-00-01290-4.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/s0025-5718-00-01290-4","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-00-01290-4","pdf_url":"https://www.ams.org/mcom/2001-70-236/S0025-5718-00-01290-4/S0025-5718-00-01290-4.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/W1972837770.pdf","grobid_xml":"https://content.openalex.org/works/W1972837770.grobid-xml"},"referenced_works_count":18,"referenced_works":["https://openalex.org/W1497177890","https://openalex.org/W1500697312","https://openalex.org/W1597196458","https://openalex.org/W1995950288","https://openalex.org/W2011093758","https://openalex.org/W2015669348","https://openalex.org/W2021498897","https://openalex.org/W2024535077","https://openalex.org/W2028841805","https://openalex.org/W2048513764","https://openalex.org/W2074663673","https://openalex.org/W2578639830","https://openalex.org/W4240859510","https://openalex.org/W4246279429","https://openalex.org/W4249883179","https://openalex.org/W4250221844","https://openalex.org/W4254911770","https://openalex.org/W4296300736"],"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/W2478288626","https://openalex.org/W2350741829"],"abstract_inverted_index":{"Using":[0],"a":[1,91,95],"carefully":[2],"optimized":[3],"segmented":[4],"sieve":[5],"and":[6,17,205],"an":[7],"efficient":[8],"checking":[9],"algorithm,":[10],"the":[11,67,83,87,197,209],"Goldbach":[12],"conjecture":[13],"has":[14],"been":[15],"verified":[16],"is":[18,94,206],"now":[19],"known":[20],"to":[21,25,57,202],"be":[22,203],"true":[23],"up":[24],"<inline-formula":[26,70,97,126,156,183],"content-type=\"math/mathml\">":[27,71,98,127,157,184],"<mml:math":[28,72,99,128,158,185],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[29,73,100,129,159,186],"alttext=\"4":[30],"dot":[31],"10":[32],"Superscript":[33],"14\">":[34],"<mml:semantics>":[35,75,107,135,164,188],"<mml:mrow>":[36,108,136,165],"<mml:mn>4</mml:mn>":[37],"<mml:mo>\u22c5</mml:mo>":[38],"<mml:msup>":[39,140,166],"<mml:mn>10</mml:mn>":[40],"<mml:mrow":[41],"class=\"MJX-TeXAtom-ORD\">":[42],"<mml:mn>14</mml:mn>":[43],"</mml:mrow>":[44,46,115,144,172],"</mml:msup>":[45,143,169],"<mml:annotation":[47,77,116,145,173,190],"encoding=\"application/x-tex\">4\\cdot":[48],"10^{14}</mml:annotation>":[49],"</mml:semantics>":[50,79,122,149,177,192],"</mml:math>":[51,80,123,150,178,193],"</inline-formula>.":[52,179],"The":[53,180],"program":[54],"was":[55,200],"distributed":[56],"various":[58],"workstations.":[59],"It":[60],"kept":[61],"track":[62],"of":[63,66,86],"maximal":[64,181],"values":[65],"smaller":[68],"prime":[69,161,182],"alttext=\"p\">":[74,187],"<mml:mi>p</mml:mi>":[76,112,141,167,171,189],"encoding=\"application/x-tex\">p</mml:annotation>":[78,191],"</inline-formula>":[81,124,151,194],"in":[82,196],"minimal":[84,92],"partition":[85,93,210],"even":[88],"numbers,":[89],"where":[90],"representation":[96],"alttext=\"2":[101,130],"n":[102,131],"equals":[103],"p":[104,119,133],"plus":[105],"q\">":[106],"<mml:mn>2</mml:mn>":[109,137],"<mml:mi>n</mml:mi>":[110,138],"<mml:mo>=</mml:mo>":[111],"<mml:mo>+</mml:mo>":[113],"<mml:mi>q</mml:mi>":[114],"encoding=\"application/x-tex\">2n":[117,146],"=":[118,212],"+":[120,214],"q</mml:annotation>":[121],"with":[125],"minus":[132],"prime\">":[134],"<mml:mo>\u2212</mml:mo>":[139],"<mml:mo>\u2032</mml:mo>":[142,168],"-":[147],"p\u2019</mml:annotation>":[148],"being":[152],"composite":[153],"for":[154,208],"all":[155],"alttext=\"p":[160],"greater-than":[162],"p\">":[163],"<mml:mo>&gt;</mml:mo>":[170],"encoding=\"application/x-tex\">p\u2019":[174],"&gt;":[175],"p</mml:annotation>":[176],"needed":[195,207],"considered":[198],"interval":[199],"found":[201],"5569":[204,213],"389965026819938":[211],"389965026814369.":[215]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":3},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":5},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":4},{"year":2017,"cited_by_count":4},{"year":2016,"cited_by_count":3},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":11},{"year":2012,"cited_by_count":5}],"updated_date":"2026-04-23T09:07:50.710637","created_date":"2025-10-10T00:00:00"}