{"id":"https://openalex.org/W7225464974","doi":"https://doi.org/10.5281/zenodo.24034784","title":"The Beal Conjecture: A Structure-First Investigation, ...","display_name":"The Beal Conjecture: A Structure-First Investigation, ...","publication_year":2026,"publication_date":"2026-11-07","ids":{"openalex":"https://openalex.org/W7225464974","doi":"https://doi.org/10.5281/zenodo.24034784"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.24034784","doi":"https://doi.org/10.5281/zenodo.24034784","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24034784","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.24034784","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5036262330","display_name":"C. Tantisukarom","orcid":"https://orcid.org/0009-0002-5423-5561","observed_orcids":["https://orcid.org/0009-0002-5423-5561"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Tantisukarom, Chaiya","raw_affiliation_strings":[],"raw_orcid":"https://orcid.org/0009-0002-5423-5561","affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5036262330"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"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":0.9911999702453613,"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":0.9911999702453613,"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/T11166","display_name":"Analytic Number Theory Research","score":0.0066999997943639755,"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":0.00039999998989515007,"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/beal-s-conjecture","display_name":"Beal's conjecture","score":0.949999988079071},{"id":"https://openalex.org/keywords/fermat-s-last-theorem","display_name":"Fermat's Last Theorem","score":0.8330000042915344},{"id":"https://openalex.org/keywords/frey-curves","display_name":"Frey curves","score":0.828000009059906},{"id":"https://openalex.org/keywords/diophantine-equations","display_name":"Diophantine equations","score":0.8140000104904175},{"id":"https://openalex.org/keywords/common-prime-factor","display_name":"common prime factor","score":0.7250000238418579},{"id":"https://openalex.org/keywords/exponents","display_name":"exponents","score":0.4230000078678131}],"concepts":[{"id":"https://openalex.org/C2777179996","wikidata":"https://www.wikidata.org/wiki/Q911222","display_name":"Mistake","level":2,"score":0.7203999757766724},{"id":"https://openalex.org/C2777026412","wikidata":"https://www.wikidata.org/wiki/Q2684591","display_name":"Statement (logic)","level":2,"score":0.5914000272750854},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5669000148773193},{"id":"https://openalex.org/C169654258","wikidata":"https://www.wikidata.org/wiki/Q12479","display_name":"Number theory","level":2,"score":0.4406000077724457},{"id":"https://openalex.org/C188139237","wikidata":"https://www.wikidata.org/wiki/Q50706","display_name":"Square-free integer","level":2,"score":0.42010000348091125},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.4172999858856201},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3968000113964081},{"id":"https://openalex.org/C135692309","wikidata":"https://www.wikidata.org/wiki/Q111124","display_name":"Square (algebra)","level":2,"score":0.36970001459121704},{"id":"https://openalex.org/C48103436","wikidata":"https://www.wikidata.org/wiki/Q599031","display_name":"State (computer science)","level":2,"score":0.3668999969959259},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.31540000438690186},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.305400013923645},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.29490000009536743},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.2874999940395355},{"id":"https://openalex.org/C113429393","wikidata":"https://www.wikidata.org/wiki/Q49008","display_name":"Prime number","level":2,"score":0.2689000070095062},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.25949999690055847},{"id":"https://openalex.org/C162838799","wikidata":"https://www.wikidata.org/wiki/Q596077","display_name":"Counterexample","level":2,"score":0.25220000743865967}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.24034784","doi":"https://doi.org/10.5281/zenodo.24034784","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24034784","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},{"id":"doi:10.5281/zenodo.24034785","doi":"https://doi.org/10.5281/zenodo.24034785","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24034785","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.5281/zenodo.24034784","doi":"https://doi.org/10.5281/zenodo.24034784","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24034784","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7225468228","https://openalex.org/W7225144666","https://openalex.org/W6902278050","https://openalex.org/W6949435919","https://openalex.org/W7137009074","https://openalex.org/W7155521213","https://openalex.org/W6949706502","https://openalex.org/W2906658995","https://openalex.org/W6939686158","https://openalex.org/W7131379186"],"abstract_inverted_index":{"This":[0],"document":[1,141],"consolidates":[2],"a":[3,23,39,78,103,146],"long":[4],"working":[5],"discussion":[6,27,148],"of":[7,50,80,94,134],"the":[8,33,44,66,74,91,108,129,135],"Beal":[9,136],"conjecture:":[10],"if":[11],"$A^x+B^y=C^z$":[12],"with":[13,97],"$A,B,C$":[14,21],"positive":[15],"integers":[16],"and":[17,42,57,89,126],"$x,y,z\\ge":[18],"3$,":[19],"then":[20],"share":[22],"prime":[24],"factor.":[25],"The":[26,140],"followed":[28],"one":[29],"methodological":[30],"principle:":[31],"treat":[32],"six":[34],"free":[35,81],"quantities":[36],"$(A,B,C,x,y,z)$":[37],"through":[38],"single":[40],"structure,":[41,85],"explain":[43],"smallest":[45],"instance":[46],"($3^3+6^3=3^5$)":[47],"reproducibly":[48],"instead":[49],"searching.":[51],"We":[52],"record":[53],"(i)":[54],"exact":[55],"reductions":[56],"equivalent":[58],"formulations,":[59],"(ii)":[60],"case":[61],"studies":[62],"($(3,3,3)$,":[63],"$(3,3,4)$,":[64],"$(3,3,r)$,":[65],"square":[67],"term,":[68],"three":[69],"distinct":[70],"odd":[71],"primes),":[72],"(iii)":[73],"Catalan--Mihailescu":[75],"proof":[76,133],"as":[77],"model":[79],"exponents":[82],"tamed":[83],"by":[84],"(iv)":[86],"Frey-curve":[87],"face-control,":[88],"(v)":[90],"current":[92],"state":[93],"Darmon's":[95],"program,":[96],"literature":[98],"citations.":[99],"Every":[100],"statement":[101],"carries":[102],"status":[104],"tag":[105],"(proved":[106],"in":[107],"literature,":[109],"derived":[110],"here,":[111,113],"computed":[112],"heuristic,":[114],"or":[115,153],"recalled":[116],"from":[117],"memory).":[118],"A":[119],"corrections":[120],"log":[121],"lists":[122],"every":[123],"mistake":[124],"made":[125],"retracted":[127],"during":[128],"discussion.":[130],"\\emph{No":[131],"new":[132,147],"conjecture":[137],"is":[138,142],"claimed.}":[139],"written":[143],"so":[144],"that":[145],"can":[149],"resume":[150],"without":[151],"re-deriving":[152],"re-verifying":[154],"anything.":[155]},"counts_by_year":[],"updated_date":"2026-11-09T13:22:03.289127","created_date":"2026-11-08T00:00:00"}