{"id":"https://openalex.org/W2309335899","doi":"https://doi.org/10.1090/mcom/3129","title":"Sums of two \ud835\udc46-units via Frey-Hellegouarch curves","display_name":"Sums of two \ud835\udc46-units via Frey-Hellegouarch curves","publication_year":2016,"publication_date":"2016-07-14","ids":{"openalex":"https://openalex.org/W2309335899","doi":"https://doi.org/10.1090/mcom/3129","mag":"2309335899"},"language":"en","primary_location":{"id":"doi:10.1090/mcom/3129","is_oa":false,"landing_page_url":"https://doi.org/10.1090/mcom/3129","pdf_url":null,"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":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1603.07922","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5102805894","display_name":"Michael A. Bennett","orcid":"https://orcid.org/0000-0002-8888-6218"},"institutions":[{"id":"https://openalex.org/I141945490","display_name":"University of British Columbia","ror":"https://ror.org/03rmrcq20","country_code":"CA","type":"education","lineage":["https://openalex.org/I141945490"]}],"countries":["CA"],"is_corresponding":true,"raw_author_name":"Michael Bennett","raw_affiliation_strings":["Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada","UBC Mathematics - Department of Mathematics [Vancouver] (Faculty of Science, University of British Columbia,1984 Mathematics Road, Vancouver, BC V6T 1Z2 - Canada)"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada","institution_ids":["https://openalex.org/I141945490"]},{"raw_affiliation_string":"UBC Mathematics - Department of Mathematics [Vancouver] (Faculty of Science, University of British Columbia,1984 Mathematics Road, Vancouver, BC V6T 1Z2 - Canada)","institution_ids":["https://openalex.org/I141945490"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5112871705","display_name":"Nicolas Billerey","orcid":null},"institutions":[{"id":"https://openalex.org/I1294671590","display_name":"Centre National de la Recherche Scientifique","ror":"https://ror.org/02feahw73","country_code":"FR","type":"government","lineage":["https://openalex.org/I1294671590"]},{"id":"https://openalex.org/I198244214","display_name":"Universit\u00e9 Clermont Auvergne","ror":"https://ror.org/01a8ajp46","country_code":"FR","type":"education","lineage":["https://openalex.org/I198244214"]},{"id":"https://openalex.org/I4210155549","display_name":"Laboratoire de Math\u00e9matiques Blaise Pascal","ror":"https://ror.org/05sd5r855","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I1294671590","https://openalex.org/I198244214","https://openalex.org/I4210141950","https://openalex.org/I4210155549"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Nicolas Billerey","raw_affiliation_strings":["Laboratoire de Math\u00c3\u00a9matiques, Universit\u00c3\u00a9 Clermont Auvergne, Universit\u00c3\u00a9 Blaise Pascal, BP 10448, F-63000 Clermont-Ferrand, France \u00e2\u0080\u0094 and \u00e2\u0080\u0094 CNRS, UMR 6620, LM, F-63171 Aubi\u00c3\u00a8re, France","LMBP - Laboratoire de Math\u00e9matiques Blaise Pascal (Campus universitaire des C\u00e9zeaux, 3 place Vasarely, TSA 60026 / CS 60026, 63178 Aubi\u00e8re Cedex - France)"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Laboratoire de Math\u00c3\u00a9matiques, Universit\u00c3\u00a9 Clermont Auvergne, Universit\u00c3\u00a9 Blaise Pascal, BP 10448, F-63000 Clermont-Ferrand, France \u00e2\u0080\u0094 and \u00e2\u0080\u0094 CNRS, UMR 6620, LM, F-63171 Aubi\u00c3\u00a8re, France","institution_ids":["https://openalex.org/I198244214","https://openalex.org/I1294671590"]},{"raw_affiliation_string":"LMBP - Laboratoire de Math\u00e9matiques Blaise Pascal (Campus universitaire des C\u00e9zeaux, 3 place Vasarely, TSA 60026 / CS 60026, 63178 Aubi\u00e8re Cedex - France)","institution_ids":["https://openalex.org/I4210155549"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5102805894"],"corresponding_institution_ids":["https://openalex.org/I141945490"],"apc_list":null,"apc_paid":null,"fwci":1.0653,"has_fulltext":false,"cited_by_count":9,"citation_normalized_percentile":{"value":0.75582093,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"86","issue":"305","first_page":"1375","last_page":"1401"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.9995999932289124,"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"}},"topics":[{"id":"https://openalex.org/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.9995999932289124,"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"}},{"id":"https://openalex.org/T11435","display_name":"Polynomial and algebraic computation","score":0.9782000184059143,"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/T11693","display_name":"Cryptography and Residue Arithmetic","score":0.9717000126838684,"subfield":{"id":"https://openalex.org/subfields/1710","display_name":"Information Systems"},"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":0.9072551131248474},{"id":"https://openalex.org/keywords/modularity","display_name":"Modularity (biology)","score":0.5948879718780518},{"id":"https://openalex.org/keywords/carry","display_name":"Carry (investment)","score":0.5926277041435242},{"id":"https://openalex.org/keywords/logarithm","display_name":"Logarithm","score":0.5629880428314209},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.5016896724700928},{"id":"https://openalex.org/keywords/galois-module","display_name":"Galois module","score":0.4672296941280365},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.4562569260597229},{"id":"https://openalex.org/keywords/perfect-power","display_name":"Perfect power","score":0.43774205446243286},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.42124488949775696},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.0835777223110199},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.08257463574409485}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.9072551131248474},{"id":"https://openalex.org/C2779478453","wikidata":"https://www.wikidata.org/wiki/Q6889748","display_name":"Modularity (biology)","level":2,"score":0.5948879718780518},{"id":"https://openalex.org/C2776299755","wikidata":"https://www.wikidata.org/wiki/Q432449","display_name":"Carry (investment)","level":2,"score":0.5926277041435242},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.5629880428314209},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.5016896724700928},{"id":"https://openalex.org/C145899342","wikidata":"https://www.wikidata.org/wiki/Q3427418","display_name":"Galois module","level":2,"score":0.4672296941280365},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.4562569260597229},{"id":"https://openalex.org/C93446932","wikidata":"https://www.wikidata.org/wiki/Q1972803","display_name":"Perfect power","level":2,"score":0.43774205446243286},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.42124488949775696},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0835777223110199},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.08257463574409485},{"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/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C54355233","wikidata":"https://www.wikidata.org/wiki/Q7162","display_name":"Genetics","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1090/mcom/3129","is_oa":false,"landing_page_url":"https://doi.org/10.1090/mcom/3129","pdf_url":null,"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:arXiv.org:1603.07922","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1603.07922","pdf_url":"https://arxiv.org/pdf/1603.07922","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"text"},{"id":"pmh:oai:HAL:hal-01292805v2","is_oa":false,"landing_page_url":"https://hal.science/hal-01292805","pdf_url":null,"source":{"id":"https://openalex.org/S4306402512","display_name":"HAL (Le Centre pour la Communication Scientifique Directe)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1294671590","host_organization_name":"Centre National de la Recherche Scientifique","host_organization_lineage":["https://openalex.org/I1294671590"],"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, 2016, 86 (305), pp.1375-1401. &#x27E8;10.1090/mcom/3129&#x27E9;","raw_type":"Journal articles"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1603.07922","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1603.07922","pdf_url":"https://arxiv.org/pdf/1603.07922","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"text"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G6515991890","display_name":null,"funder_award_id":"ANR-14-CE25-0015","funder_id":"https://openalex.org/F4320320883","funder_display_name":"Agence Nationale de la Recherche"}],"funders":[{"id":"https://openalex.org/F4320320883","display_name":"Agence Nationale de la Recherche","ror":"https://ror.org/00rbzpz17"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":24,"referenced_works":["https://openalex.org/W131263910","https://openalex.org/W1486898453","https://openalex.org/W1490007780","https://openalex.org/W1527292340","https://openalex.org/W1570844510","https://openalex.org/W1868013030","https://openalex.org/W1970456472","https://openalex.org/W1976677460","https://openalex.org/W1998239757","https://openalex.org/W2006077852","https://openalex.org/W2029704363","https://openalex.org/W2045396152","https://openalex.org/W2084490540","https://openalex.org/W2115500455","https://openalex.org/W2137844544","https://openalex.org/W2142966800","https://openalex.org/W2166349036","https://openalex.org/W2314098110","https://openalex.org/W2323536774","https://openalex.org/W2326616179","https://openalex.org/W2528471897","https://openalex.org/W2963535644","https://openalex.org/W3101951724","https://openalex.org/W4292027090"],"related_works":["https://openalex.org/W4230687177","https://openalex.org/W4229867111","https://openalex.org/W1968829728","https://openalex.org/W1488770949","https://openalex.org/W2349859869","https://openalex.org/W2793180915","https://openalex.org/W3120118008","https://openalex.org/W4316651471","https://openalex.org/W2051365847","https://openalex.org/W2156551154"],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"we":[3,119],"develop":[4],"a":[5,52,94],"new":[6],"method":[7],"for":[8,68,77,99,122],"finding":[9],"all":[10],"perfect":[11],"powers":[12],"which":[13],"can":[14],"be":[15],"expressed":[16],"as":[17],"the":[18,62,69],"sum":[19],"of":[20,55,64,103],"two":[21],"rational":[22],"<inline-formula":[23,38,105,123,160],"content-type=\"math/mathml\">":[24,39,106,124,161],"<mml:math":[25,40,107,125,162],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[26,41,108,126,163],"alttext=\"upper":[27,42,109,127,164],"S\">":[28,43,110],"<mml:semantics>":[29,44,111,135,174],"<mml:mi>S</mml:mi>":[30,45,112,137,176],"<mml:annotation":[31,46,113,149,190],"encoding=\"application/x-tex\">S</mml:annotation>":[32,47,114],"</mml:semantics>":[33,48,115,156,197],"</mml:math>":[34,49,116,157,198],"</inline-formula>":[35,50,117,158,199],"-units,":[36],"where":[37],"is":[51,59,85],"finite":[53],"set":[54],"primes.":[56],"Our":[57],"approach":[58],"based":[60],"upon":[61],"modularity":[63],"Galois":[65],"representations":[66],"and,":[67],"most":[70],"part,":[71],"does":[72],"not":[73],"require":[74],"lower":[75],"bounds":[76],"linear":[78],"forms":[79],"in":[80],"logarithms.":[81],"Its":[82],"main":[83],"virtue":[84],"that":[86],"it":[87],"enables":[88],"us":[89],"to":[90],"carry":[91],"out":[92],"such":[93],"program":[95],"explicitly,":[96],"at":[97],"least":[98],"certain":[100],"small":[101],"sets":[102],"primes":[104],";":[118],"do":[120],"so":[121],"S":[128,165],"equals":[129,166],"StartSet":[130,167],"2":[131],"comma":[132,169,171],"3":[133,154,168],"EndSet\">":[134,173],"<mml:mrow>":[136,175],"<mml:mo>=</mml:mo>":[138,177],"<mml:mo":[139,145,178,186],"fence=\"false\"":[140,146,179,187],"stretchy=\"false\">{</mml:mo>":[141,180],"<mml:mn>2</mml:mn>":[142],"<mml:mo>,</mml:mo>":[143,182,184],"<mml:mn>3</mml:mn>":[144,181],"stretchy=\"false\">}</mml:mo>":[147,188],"</mml:mrow>":[148,189],"encoding=\"application/x-tex\">S":[150],"=":[151],"\\{":[152,192],"2,":[153],"\\}</mml:annotation>":[155,196],"and":[159],"5":[170],"7":[172,195],"<mml:mn>5</mml:mn>":[183],"<mml:mn>7</mml:mn>":[185],"encoding=\"application/x-tex\">S=":[191],"3,":[193],"5,":[194],".":[200]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2016,"cited_by_count":1}],"updated_date":"2026-05-21T09:19:25.381259","created_date":"2025-10-10T00:00:00"}