{"id":"https://openalex.org/W123232278","doi":"https://doi.org/10.1090/s0025-5718-99-01067-4","title":"Nonexistence conditions of a solution for the congruence \ud835\udc65\u2081^{\ud835\udc58}+\u22ef+\ud835\udc65_{\ud835\udc60}^{\ud835\udc58}\u2261\ud835\udc41(\\mod\ud835\udc5d\u207f)","display_name":"Nonexistence conditions of a solution for the congruence \ud835\udc65\u2081^{\ud835\udc58}+\u22ef+\ud835\udc65_{\ud835\udc60}^{\ud835\udc58}\u2261\ud835\udc41(\\mod\ud835\udc5d\u207f)","publication_year":1999,"publication_date":"1999-02-24","ids":{"openalex":"https://openalex.org/W123232278","doi":"https://doi.org/10.1090/s0025-5718-99-01067-4","mag":"123232278"},"language":"lv","primary_location":{"id":"doi:10.1090/s0025-5718-99-01067-4","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-99-01067-4","pdf_url":"https://www.ams.org/mcom/1999-68-227/S0025-5718-99-01067-4/S0025-5718-99-01067-4.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/1999-68-227/S0025-5718-99-01067-4/S0025-5718-99-01067-4.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5021311433","display_name":"Hiroshi Sekigawa","orcid":"https://orcid.org/0009-0003-1770-0994"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Hiroshi Sekigawa","raw_affiliation_strings":["NTT Communication Science Laboratories, 2-4 Hikaridai, Seika-cho, Soraku-gun Kyoto 619-0237 Japan","NTT Communication Science Labs, 2-4 Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-0237, Japan#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"NTT Communication Science Laboratories, 2-4 Hikaridai, Seika-cho, Soraku-gun Kyoto 619-0237 Japan","institution_ids":[]},{"raw_affiliation_string":"NTT Communication Science Labs, 2-4 Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-0237, Japan#TAB#","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5010911834","display_name":"Kenji Koyama","orcid":"https://orcid.org/0000-0002-1467-7050"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Kenji Koyama","raw_affiliation_strings":["NTT Communication Science Laboratories, 2-4 Hikaridai, Seika-cho, Soraku-gun Kyoto 619-0237 Japan","NTT Communication Science Labs, 2-4 Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-0237, Japan#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"NTT Communication Science Laboratories, 2-4 Hikaridai, Seika-cho, Soraku-gun Kyoto 619-0237 Japan","institution_ids":[]},{"raw_affiliation_string":"NTT Communication Science Labs, 2-4 Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-0237, Japan#TAB#","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.7452,"has_fulltext":true,"cited_by_count":4,"citation_normalized_percentile":{"value":0.57647059,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"68","issue":"227","first_page":"1283","last_page":"1297"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11166","display_name":"Analytic Number Theory Research","score":0.9987999796867371,"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":0.9987999796867371,"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/T11680","display_name":"Advanced Algebra and Geometry","score":0.9940000176429749,"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/T11476","display_name":"Graph theory and applications","score":0.9926000237464905,"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":0.5778408646583557},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.3452686667442322},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.2992165684700012}],"concepts":[{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.5778408646583557},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.3452686667442322},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.2992165684700012}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1090/s0025-5718-99-01067-4","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-99-01067-4","pdf_url":"https://www.ams.org/mcom/1999-68-227/S0025-5718-99-01067-4/S0025-5718-99-01067-4.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/s0025-5718-99-01067-4","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-99-01067-4","pdf_url":"https://www.ams.org/mcom/1999-68-227/S0025-5718-99-01067-4/S0025-5718-99-01067-4.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/W123232278.pdf","grobid_xml":"https://content.openalex.org/works/W123232278.grobid-xml"},"referenced_works_count":9,"referenced_works":["https://openalex.org/W656708819","https://openalex.org/W792717713","https://openalex.org/W983763560","https://openalex.org/W1146347781","https://openalex.org/W2015585803","https://openalex.org/W2076216031","https://openalex.org/W2082811707","https://openalex.org/W2329964106","https://openalex.org/W2798483379"],"related_works":["https://openalex.org/W2051487156","https://openalex.org/W2073681303","https://openalex.org/W2544423928","https://openalex.org/W2052122378","https://openalex.org/W2053286651","https://openalex.org/W2181743346","https://openalex.org/W2187401768","https://openalex.org/W2181413294","https://openalex.org/W2989452537","https://openalex.org/W4313731559"],"abstract_inverted_index":{"We":[0,165],"obtain":[1],"nonexistence":[2,167],"conditions":[3,168],"of":[4,8,169,287],"a":[5,162],"solution":[6],"for":[7,214,321],"the":[9,170,288],"congruence":[10],"<inline-formula":[11,84,106,128,144,172,215,234,247,260,273,290,322],"content-type=\"math/mathml\">":[12,85,107,129,145,173,216,235,248,261,274,291,323],"<mml:math":[13,86,108,130,146,174,217,236,249,262,275,292,324],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[14,87,109,131,147,175,218,237,250,263,276,293,325],"alttext=\"x":[15],"1":[16],"Superscript":[17,26,35,149,183,298],"k":[18,27],"Baseline":[19,28,37,185,300],"plus":[20,22],"midline-horizontal-ellipsis":[21],"x":[23],"Subscript":[24],"s":[25,177,295],"identical-to":[29],"upper":[30,179],"N":[31,180],"left-parenthesis":[32],"mod":[33,181],"p":[34,182,297],"n":[36,184,299],"right-parenthesis\">":[38,186,301],"<mml:semantics>":[39,91,113,134,151,187,222,239,252,265,278,302,329],"<mml:mrow>":[40,92,114,188,223,303,330],"<mml:msubsup>":[41,51],"<mml:mi>x</mml:mi>":[42,52],"<mml:mn>1</mml:mn>":[43],"<mml:mi>k</mml:mi>":[44,54,93,224,331],"</mml:msubsup>":[45,55],"<mml:mo>+</mml:mo>":[46,50],"<mml:mo>":[47,56,94,116],"\u22ef":[48],"</mml:mo>":[49,58,96,118],"<mml:mi>s</mml:mi>":[53,115,191,306],"\u2261":[57],"<mml:mi>N</mml:mi>":[59,135,193],"<mml:mspace":[60,65],"width=\"0.667em\"/>":[61],"<mml:mo":[62,71,189,194,204,304,312],"stretchy=\"false\">(</mml:mo>":[63,190,305],"<mml:mi>mod</mml:mi>":[64],"width=\"0.333em\"/>":[66],"<mml:msup>":[67,152,199,308],"<mml:mi>p</mml:mi>":[68,153,200,309],"<mml:mi>n</mml:mi>":[69,154,201,310],"</mml:msup>":[70,155,202,311],"stretchy=\"false\">)</mml:mo>":[72,205,313],"</mml:mrow>":[73,98,120,203,206,227,314],"<mml:annotation":[74,99,121,136,156,207,228,241,254,267,280,315],"encoding=\"application/x-tex\">x_1^k+\\cdots":[75],"+x_s^k\\equiv":[76],"N\\pmod":[77],"{p^n}</mml:annotation>":[78],"</mml:semantics>":[79,102,124,138,158,211,230,243,256,269,282,318],"</mml:math>":[80,103,125,139,159,212,231,244,257,270,283,319],"</inline-formula>":[81,104,126,140,160,213,232,245,258,271,284,320],",":[82,105,233,246,259,272,285],"where":[83],"alttext=\"k":[88,219,326],"greater-than-or-equal-to":[89,111],"2\">":[90,112,221],"\u2265":[95,117],"<mml:mn>2</mml:mn>":[97,119,226],"encoding=\"application/x-tex\">k\\geq":[100],"2</mml:annotation>":[101,123],"alttext=\"s":[110],"encoding=\"application/x-tex\">s\\geq":[122],"and":[127,143,286],"alttext=\"upper":[132],"N\">":[133],"encoding=\"application/x-tex\">N</mml:annotation>":[137],"are":[141],"integers,":[142],"alttext=\"p":[148],"n\">":[150],"encoding=\"application/x-tex\">p^n</mml:annotation>":[157],"is":[161],"prime":[163],"power.":[164],"give":[166],"form":[171,289],"alttext=\"left-parenthesis":[176,294],"comma":[178,296],"<mml:mo>,</mml:mo>":[192,307],"lspace=\"thickmathspace\"":[195],"rspace=\"thickmathspace\">mod</mml:mo>":[196],"<mml:mrow":[197],"class=\"MJX-TeXAtom-ORD\">":[198],"encoding=\"application/x-tex\">(s,":[208,316],"N\\bmod":[209],"{p^n})</mml:annotation>":[210],"equals":[220,327],"<mml:mo>=</mml:mo>":[225,332],"encoding=\"application/x-tex\">k=2</mml:annotation>":[229],"alttext=\"3\">":[238],"<mml:mn>3</mml:mn>":[240],"encoding=\"application/x-tex\">3</mml:annotation>":[242],"alttext=\"4\">":[251],"<mml:mn>4</mml:mn>":[253],"encoding=\"application/x-tex\">4</mml:annotation>":[255],"alttext=\"5\">":[264],"<mml:mn>5</mml:mn>":[266],"encoding=\"application/x-tex\">5</mml:annotation>":[268],"alttext=\"7\">":[277],"<mml:mn>7</mml:mn>":[279],"encoding=\"application/x-tex\">7</mml:annotation>":[281],"p^n)</mml:annotation>":[317],"11\">":[328],"<mml:mn>11</mml:mn>":[333]},"counts_by_year":[{"year":2022,"cited_by_count":1}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}
