{"id":"https://openalex.org/W7214715866","doi":"https://doi.org/10.48550/arxiv.2609.32509","title":"On Fermat-type equations of signature $(r,r,p)$","display_name":"On Fermat-type equations of signature $(r,r,p)$","publication_year":2026,"publication_date":"2026-09-26","ids":{"openalex":"https://openalex.org/W7214715866","doi":"https://doi.org/10.48550/arxiv.2609.32509"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.32509","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.32509","pdf_url":"https://arxiv.org/pdf/2609.32509","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.32509","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5012345793","display_name":"Nuno Freitas","orcid":"https://orcid.org/0000-0002-4740-3427","observed_orcids":["https://orcid.org/0000-0002-4740-3427"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Freitas, Nuno","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5012345793"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.9648000001907349,"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":0.9648000001907349,"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/T11680","display_name":"Advanced Algebra and Geometry","score":0.020600000396370888,"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/T11166","display_name":"Analytic Number Theory Research","score":0.007600000128149986,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/fermat-type-equations","display_name":"Fermat-type equations","score":0.9829999804496765},{"id":"https://openalex.org/keywords/diophantine-equations","display_name":"Diophantine equations","score":0.75},{"id":"https://openalex.org/keywords/frey-curves","display_name":"Frey curves","score":0.777999997138977},{"id":"https://openalex.org/keywords/frobenius-traces","display_name":"trace of Frobenius","score":0.7599999904632568},{"id":"https://openalex.org/keywords/level-lowering","display_name":"level lowering","score":0.6790000200271606},{"id":"https://openalex.org/keywords/primitive-solutions","display_name":"primitive solutions","score":0.8019999861717224},{"id":"https://openalex.org/keywords/modular-elliptic-curves","display_name":"modular elliptic curves","score":0.5379999876022339}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7685999870300293},{"id":"https://openalex.org/C75291252","wikidata":"https://www.wikidata.org/wiki/Q1315756","display_name":"TRACE (psycholinguistics)","level":2,"score":0.6985999941825867},{"id":"https://openalex.org/C2779696439","wikidata":"https://www.wikidata.org/wiki/Q7512811","display_name":"Signature (topology)","level":2,"score":0.6758999824523926},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4341999888420105},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.41940000653266907},{"id":"https://openalex.org/C98184364","wikidata":"https://www.wikidata.org/wiki/Q1780131","display_name":"Argument (complex analysis)","level":2,"score":0.3774000108242035},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.36239999532699585},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3495999872684479},{"id":"https://openalex.org/C131220774","wikidata":"https://www.wikidata.org/wiki/Q1441456","display_name":"Ramanujan's sum","level":2,"score":0.32850000262260437},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.32670000195503235},{"id":"https://openalex.org/C167729594","wikidata":"https://www.wikidata.org/wiki/Q17736","display_name":"Axiom","level":2,"score":0.3255000114440918},{"id":"https://openalex.org/C169654258","wikidata":"https://www.wikidata.org/wiki/Q12479","display_name":"Number theory","level":2,"score":0.3255000114440918},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.27489998936653137},{"id":"https://openalex.org/C2777210771","wikidata":"https://www.wikidata.org/wiki/Q4927124","display_name":"Block (permutation group theory)","level":2,"score":0.27399998903274536},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.2639999985694885}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.32509","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.32509","pdf_url":"https://arxiv.org/pdf/2609.32509","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2609.32509","doi":"https://doi.org/10.48550/arxiv.2609.32509","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.32509","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.32509","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.32509","pdf_url":"https://arxiv.org/pdf/2609.32509","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214715866.pdf","grobid_xml":"https://content.openalex.org/works/W7214715866.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W2949453072","https://openalex.org/W4387166842","https://openalex.org/W1505449575","https://openalex.org/W4380624831","https://openalex.org/W4414790498","https://openalex.org/W4226212681","https://openalex.org/W4285428933","https://openalex.org/W2364327414","https://openalex.org/W7116854701","https://openalex.org/W4296021881"],"abstract_inverted_index":{"Let":[0],"$r\\geq11$":[1],"be":[2],"a":[3,54,68,76],"prime.":[4],"We":[5],"show":[6],"that":[7,60],"there":[8],"are":[9],"infinitely":[10],"many":[11],"integers":[12],"$C$":[13],"for":[14,27],"which":[15,67],"the":[16,51,79],"Fermat-type":[17],"equation":[18],"$$":[19,21],"x^r+y^r=Cz^p":[20],"has":[22],"no":[23],"non-trivial":[24],"primitive":[25],"solutions":[26],"all":[28],"sufficiently":[29],"large":[30],"(in":[31],"terms":[32],"of":[33],"$r$":[34],"and~$C$)":[35],"prime":[36],"exponents~$p$.":[37],"The":[38],"proof":[39],"uses":[40],"several":[41],"Frey":[42,70],"curves":[43],"to":[44],"force":[45],"simultaneous":[46],"Frobenius":[47],"trace":[48,56],"equalities":[49,62],"at":[50],"primes":[52],"above~$3$;":[53],"new":[55],"separation":[57],"argument":[58],"shows":[59],"these":[61],"imply":[63],"$3\\mid":[64],"x+y$,":[65],"after":[66],"further":[69],"curve":[71],"and":[72],"level":[73],"lowering":[74],"give":[75],"contradiction":[77],"with":[78],"Ramanujan":[80],"bound.":[81]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-09-30T00:00:00"}