{"id":"https://openalex.org/W1986546079","doi":"https://doi.org/10.1090/s0025-5718-98-00881-3","title":"A conjecture of Erd\u00f6s on 3-powerful numbers","display_name":"A conjecture of Erd\u00f6s on 3-powerful numbers","publication_year":1998,"publication_date":"1998-01-01","ids":{"openalex":"https://openalex.org/W1986546079","doi":"https://doi.org/10.1090/s0025-5718-98-00881-3","mag":"1986546079"},"language":"en","primary_location":{"id":"doi:10.1090/s0025-5718-98-00881-3","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-98-00881-3","pdf_url":"https://www.ams.org/mcom/1998-67-221/S0025-5718-98-00881-3/S0025-5718-98-00881-3.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/1998-67-221/S0025-5718-98-00881-3/S0025-5718-98-00881-3.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5019953042","display_name":"Jonathan Cohn","orcid":"https://orcid.org/0000-0001-8567-7034"},"institutions":[{"id":"https://openalex.org/I184558857","display_name":"Royal Holloway University of London","ror":"https://ror.org/04g2vpn86","country_code":"GB","type":"education","lineage":["https://openalex.org/I124357947","https://openalex.org/I184558857"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"J. Cohn","raw_affiliation_strings":["Department of Mathematics, Royal Holloway University of London, Egham, Surrey TW20 0EX, United Kingdom","Department of Mathematics, Royal Holloway University of London, Egham, Surrey, TW20 0EX, United Kingdom"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Royal Holloway University of London, Egham, Surrey TW20 0EX, United Kingdom","institution_ids":["https://openalex.org/I184558857"]},{"raw_affiliation_string":"Department of Mathematics, Royal Holloway University of London, Egham, Surrey, TW20 0EX, United Kingdom","institution_ids":["https://openalex.org/I184558857"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5019953042"],"corresponding_institution_ids":["https://openalex.org/I184558857"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":4,"citation_normalized_percentile":{"value":0.16770833,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"67","issue":"221","first_page":"439","last_page":"440"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11166","display_name":"Analytic Number Theory Research","score":0.9994999766349792,"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":0.9994999766349792,"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/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.9976999759674072,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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":0.9954000115394592,"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/algorithm","display_name":"Algorithm","score":0.641014039516449},{"id":"https://openalex.org/keywords/annotation","display_name":"Annotation","score":0.47662127017974854},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.42293843626976013},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.3766838312149048}],"concepts":[{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.641014039516449},{"id":"https://openalex.org/C2776321320","wikidata":"https://www.wikidata.org/wiki/Q857525","display_name":"Annotation","level":2,"score":0.47662127017974854},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.42293843626976013},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.3766838312149048}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1090/s0025-5718-98-00881-3","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-98-00881-3","pdf_url":"https://www.ams.org/mcom/1998-67-221/S0025-5718-98-00881-3/S0025-5718-98-00881-3.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:CiteSeerX.psu:10.1.1.192.3553","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.192.3553","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.ams.org/journals/mcom/1998-67-221/S0025-5718-98-00881-3/S0025-5718-98-00881-3.pdf","raw_type":"text"}],"best_oa_location":{"id":"doi:10.1090/s0025-5718-98-00881-3","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-98-00881-3","pdf_url":"https://www.ams.org/mcom/1998-67-221/S0025-5718-98-00881-3/S0025-5718-98-00881-3.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":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1986546079.pdf","grobid_xml":"https://content.openalex.org/works/W1986546079.grobid-xml"},"referenced_works_count":2,"referenced_works":["https://openalex.org/W1994710381","https://openalex.org/W2798483379"],"related_works":["https://openalex.org/W2899084033","https://openalex.org/W2748952813","https://openalex.org/W2361861616","https://openalex.org/W2263699433","https://openalex.org/W2377979023","https://openalex.org/W2218034408","https://openalex.org/W2392921965","https://openalex.org/W2358755282","https://openalex.org/W2625833328","https://openalex.org/W1533177136"],"abstract_inverted_index":{"Erd\u00f6s":[0],"conjectured":[1],"that":[2,117],"the":[3,114],"Diophantine":[4],"equation":[5],"<inline-formula":[6,40,54,76,126,139,152],"content-type=\"math/mathml\">":[7,41,55,77,127,140,153],"<mml:math":[8,42,56,78,128,141,154],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[9,43,57,79,129,142,155],"alttext=\"x":[10],"plus":[11],"y":[12],"equals":[13],"z\">":[14],"<mml:semantics>":[15,45,61,84,131,144,157],"<mml:mrow>":[16,62,85],"<mml:mi>x</mml:mi>":[17,132],"<mml:mo>+</mml:mo>":[18],"<mml:mi>y</mml:mi>":[19,145],"<mml:mo>=</mml:mo>":[20],"<mml:mi>z</mml:mi>":[21,158],"</mml:mrow>":[22,68,94],"<mml:annotation":[23,47,69,95,133,146,159],"encoding=\"application/x-tex\">x+y=z</mml:annotation>":[24],"</mml:semantics>":[25,49,72,98,135,148,161],"</mml:math>":[26,50,73,99,136,149,162],"</inline-formula>":[27,51,74,100,137,150,163],"has":[28],"infinitely":[29,122],"many":[30],"solutions":[31],"in":[32],"pairwise":[33],"coprime":[34],"3-powerful":[35],"integers,":[36],"i.e.,":[37],"positive":[38],"integers":[39,125],"alttext=\"n\">":[44],"<mml:mi>n</mml:mi>":[46,67,93],"encoding=\"application/x-tex\">n</mml:annotation>":[48],"for":[52],"which":[53,166],"alttext=\"p":[58,80],"bar":[59,82],"n\">":[60,83],"<mml:mi>p</mml:mi>":[63,87],"<mml:mo>":[64,90],"\u2223":[65,91],"</mml:mo>":[66,92],"encoding=\"application/x-tex\">p\\mid":[70],"n</mml:annotation>":[71,97],"implies":[75],"cubed":[81],"<mml:msup>":[86],"<mml:mn>3</mml:mn>":[88],"</mml:msup>":[89],"encoding=\"application/x-tex\">p^3\\mid":[96],".":[101],"This":[102,171],"was":[103,110],"recently":[104],"proved":[105],"by":[106],"Nitaj":[107],"who,":[108],"however,":[109],"unable":[111],"to":[112],"verify":[113],"further":[115],"conjecture":[116],"this":[118],"could":[119],"be":[120],"done":[121],"often":[123],"with":[124],"alttext=\"x\">":[130],"encoding=\"application/x-tex\">x</mml:annotation>":[134],",":[138],"alttext=\"y\">":[143],"encoding=\"application/x-tex\">y</mml:annotation>":[147],"and":[151],"alttext=\"z\">":[156],"encoding=\"application/x-tex\">z</mml:annotation>":[160],"none":[164],"of":[165],"is":[167,172],"a":[168],"perfect":[169],"cube.":[170],"now":[173],"demonstrated.":[174]},"counts_by_year":[{"year":2023,"cited_by_count":1}],"updated_date":"2026-04-21T06:05:55.347736","created_date":"2025-10-10T00:00:00"}