{"id":"https://openalex.org/W1996663143","doi":"https://doi.org/10.1090/s0025-5718-98-00952-1","title":"Detecting perfect powers in essentially linear time","display_name":"Detecting perfect powers in essentially linear time","publication_year":1998,"publication_date":"1998-01-01","ids":{"openalex":"https://openalex.org/W1996663143","doi":"https://doi.org/10.1090/s0025-5718-98-00952-1","mag":"1996663143"},"language":"en","primary_location":{"id":"doi:10.1090/s0025-5718-98-00952-1","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-98-00952-1","pdf_url":"https://www.ams.org/mcom/1998-67-223/S0025-5718-98-00952-1/S0025-5718-98-00952-1.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/1998-67-223/S0025-5718-98-00952-1/S0025-5718-98-00952-1.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5057606234","display_name":"Daniel Bernstein","orcid":"https://orcid.org/0000-0001-7761-5853"},"institutions":[{"id":"https://openalex.org/I39422238","display_name":"University of Illinois Chicago","ror":"https://ror.org/02mpq6x41","country_code":"US","type":"education","lineage":["https://openalex.org/I39422238"]},{"id":"https://openalex.org/I95457486","display_name":"University of California, Berkeley","ror":"https://ror.org/01an7q238","country_code":"US","type":"education","lineage":["https://openalex.org/I95457486"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Daniel Bernstein","raw_affiliation_strings":["Department of Mathematics, Statistics, and Computer Science, The University of Illinois at Chicago, Chicago, Illinois 60607\u00e2\u0080\u00937045","Department of Mathematics, University of California, Berkeley, California 94720","Department of Mathematics, Statistics, and Computer Science, The University of Illinois at Chicago, Chicago, Illinois 60607\u20137045"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Statistics, and Computer Science, The University of Illinois at Chicago, Chicago, Illinois 60607\u00e2\u0080\u00937045","institution_ids":["https://openalex.org/I39422238"]},{"raw_affiliation_string":"Department of Mathematics, University of California, Berkeley, California 94720","institution_ids":["https://openalex.org/I95457486"]},{"raw_affiliation_string":"Department of Mathematics, Statistics, and Computer Science, The University of Illinois at Chicago, Chicago, Illinois 60607\u20137045","institution_ids":["https://openalex.org/I39422238"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5057606234"],"corresponding_institution_ids":["https://openalex.org/I39422238","https://openalex.org/I95457486"],"apc_list":null,"apc_paid":null,"fwci":1.6482,"has_fulltext":true,"cited_by_count":55,"citation_normalized_percentile":{"value":0.86208681,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"67","issue":"223","first_page":"1253","last_page":"1283"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.9990000128746033,"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"}},"topics":[{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.9990000128746033,"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"}},{"id":"https://openalex.org/T11697","display_name":"Numerical Methods and Algorithms","score":0.9987000226974487,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10682","display_name":"Quantum Computing Algorithms and Architecture","score":0.9975000023841858,"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/algorithm","display_name":"Algorithm","score":0.6964008808135986},{"id":"https://openalex.org/keywords/annotation","display_name":"Annotation","score":0.5683811902999878},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.48313042521476746},{"id":"https://openalex.org/keywords/type","display_name":"Type (biology)","score":0.4739901125431061},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.34324562549591064},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.3372873067855835}],"concepts":[{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.6964008808135986},{"id":"https://openalex.org/C2776321320","wikidata":"https://www.wikidata.org/wiki/Q857525","display_name":"Annotation","level":2,"score":0.5683811902999878},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.48313042521476746},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.4739901125431061},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.34324562549591064},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.3372873067855835},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0}],"mesh":[],"locations_count":8,"locations":[{"id":"doi:10.1090/s0025-5718-98-00952-1","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-98-00952-1","pdf_url":"https://www.ams.org/mcom/1998-67-223/S0025-5718-98-00952-1/S0025-5718-98-00952-1.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:pure.tue.nl:publications/81a80bd1-0ffb-4a4b-9695-7eeee15cdc4f","is_oa":false,"landing_page_url":"https://research.tue.nl/en/publications/81a80bd1-0ffb-4a4b-9695-7eeee15cdc4f","pdf_url":null,"source":{"id":"https://openalex.org/S4406922641","display_name":"TU/e Research Portal","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Bernstein, D J 1998, 'Detecting perfect powers in essentially linear time', Mathematics of Computation, vol. 67, no. 223, pp. 1253-1283. https://doi.org/10.1090/S0025-5718-98-00952-1","raw_type":"info:eu-repo/semantics/publishedVersion"},{"id":"pmh:684227","is_oa":false,"landing_page_url":"http://library.tue.nl/csp/dare/LinkToRepository.csp?recordnumber=684227","pdf_url":null,"source":{"id":"https://openalex.org/S4406923046","display_name":"TU/e Research Portal (Eindhoven University of Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"ISSN:0025-5718","raw_type":"Article / Letter to the editor"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.11.3651","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.11.3651","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://cr.yp.to/papers/powers-ams.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.156.5024","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.156.5024","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/mcom/1998-67-223/S0025-5718-98-00952-1/S0025-5718-98-00952-1.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.20.7369","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.20.7369","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://cr.yp.to/papers/powers.ps","raw_type":"text"},{"id":"pmh:oai:library.tue.nl:684227","is_oa":false,"landing_page_url":"http://repository.tue.nl/684227","pdf_url":null,"source":{"id":"https://openalex.org/S4406923046","display_name":"TU/e Research Portal (Eindhoven University of Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"ISSN:0025-5718","raw_type":"Article / Letter to the editor"},{"id":"pmh:tue:oai:pure.tue.nl:publications/81a80bd1-0ffb-4a4b-9695-7eeee15cdc4f","is_oa":false,"landing_page_url":"https://research.tue.nl/nl/publications/81a80bd1-0ffb-4a4b-9695-7eeee15cdc4f","pdf_url":null,"source":{"id":"https://openalex.org/S4306401843","display_name":"Data Archiving and Networked Services (DANS)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1322597698","host_organization_name":"Royal Netherlands Academy of Arts and Sciences","host_organization_lineage":["https://openalex.org/I1322597698"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Mathematics of Computation, 67(223), 1253 - 1283. American Mathematical Society","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":{"id":"doi:10.1090/s0025-5718-98-00952-1","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-98-00952-1","pdf_url":"https://www.ams.org/mcom/1998-67-223/S0025-5718-98-00952-1/S0025-5718-98-00952-1.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":[{"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/W1996663143.pdf","grobid_xml":"https://content.openalex.org/works/W1996663143.grobid-xml"},"referenced_works_count":36,"referenced_works":["https://openalex.org/W391578156","https://openalex.org/W564770222","https://openalex.org/W576200119","https://openalex.org/W744813392","https://openalex.org/W1161573400","https://openalex.org/W1480319329","https://openalex.org/W1496383623","https://openalex.org/W1496801598","https://openalex.org/W1507575253","https://openalex.org/W1541596757","https://openalex.org/W1548944036","https://openalex.org/W1585954501","https://openalex.org/W1655990431","https://openalex.org/W1981663184","https://openalex.org/W1993449345","https://openalex.org/W2004203774","https://openalex.org/W2005907070","https://openalex.org/W2031855568","https://openalex.org/W2046055573","https://openalex.org/W2060520271","https://openalex.org/W2061658683","https://openalex.org/W2075488055","https://openalex.org/W2083368929","https://openalex.org/W2110640335","https://openalex.org/W2123355614","https://openalex.org/W2148861663","https://openalex.org/W2166717942","https://openalex.org/W2179025976","https://openalex.org/W2341283081","https://openalex.org/W2486762503","https://openalex.org/W2610857016","https://openalex.org/W2752853835","https://openalex.org/W3004157836","https://openalex.org/W3141719238","https://openalex.org/W4239679971","https://openalex.org/W6600623642"],"related_works":["https://openalex.org/W1892467659","https://openalex.org/W1529400504","https://openalex.org/W2349865494","https://openalex.org/W2998403542","https://openalex.org/W2808586768","https://openalex.org/W2372553222","https://openalex.org/W3157620392","https://openalex.org/W2925832130","https://openalex.org/W2045715842","https://openalex.org/W2105880240"],"abstract_inverted_index":{"This":[0],"paper":[1],"(1)":[2],"gives":[3],"complete":[4],"details":[5],"of":[6],"an":[7,30,34],"algorithm":[8,31,108],"to":[9],"compute":[10],"approximate":[11],"<inline-formula":[12,36,57,76,112],"content-type=\"math/mathml\">":[13,37,58,77,113],"<mml:math":[14,38,59,78,114],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[15,39,60,79,115],"alttext=\"k\">":[16],"<mml:semantics>":[17,43,62,81,127],"<mml:mi>k</mml:mi>":[18],"<mml:annotation":[19,49,64,83,152],"encoding=\"application/x-tex\">k</mml:annotation>":[20],"</mml:semantics>":[21,51,66,85,155],"</mml:math>":[22,52,67,86,156],"</inline-formula>":[23,53,68,87,157],"th":[24],"roots;":[25],"(2)":[26],"uses":[27],"this":[28,105],"in":[29,102,110],"that,":[32],"given":[33],"integer":[35],"alttext=\"n":[40],"greater-than":[41],"1\">":[42],"<mml:mrow>":[44,128],"<mml:mi>n</mml:mi>":[45,63,82,135],"<mml:mo>&gt;</mml:mo>":[46],"<mml:mn>1</mml:mn>":[47,141,146],"</mml:mrow>":[48,149,151],"encoding=\"application/x-tex\">n&gt;1</mml:annotation>":[50],",":[54],"either":[55],"writes":[56],"alttext=\"n\">":[61,80],"encoding=\"application/x-tex\">n</mml:annotation>":[65,84],"as":[69],"a":[70,90],"perfect":[71,91],"power":[72],"or":[73],"proves":[74],"that":[75,104],"is":[88],"not":[89],"power;":[92],"(3)":[93],"proves,":[94],"using":[95],"Loxton\u2019s":[96],"theorem":[97],"on":[98],"multiple":[99],"linear":[100],"forms":[101],"logarithms,":[103],"perfect-power":[106],"decomposition":[107],"runs":[109],"time":[111],"alttext=\"left-parenthesis":[116],"log":[117],"n":[118],"right-parenthesis":[119],"Superscript":[120],"1":[121,125],"plus":[122],"o":[123],"left-parenthesis":[124],"right-parenthesis\">":[126],"<mml:mo":[129,137,144,147],"stretchy=\"false\">(</mml:mo>":[130,145],"<mml:mi>log</mml:mi>":[131],"<mml:mo>":[132],"\u2061":[133],"</mml:mo>":[134],"<mml:msup>":[136],"stretchy=\"false\">)</mml:mo>":[138,148],"<mml:mrow":[139],"class=\"MJX-TeXAtom-ORD\">":[140],"<mml:mo>+</mml:mo>":[142],"<mml:mi>o</mml:mi>":[143],"</mml:msup>":[150],"encoding=\"application/x-tex\">(\\log":[153],"n)^{1+o(1)}</mml:annotation>":[154],".":[158]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":5},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":1},{"year":2015,"cited_by_count":1},{"year":2013,"cited_by_count":1}],"updated_date":"2026-04-21T06:05:55.347736","created_date":"2025-10-10T00:00:00"}