{"id":"https://openalex.org/W7225144666","doi":"https://doi.org/10.5281/zenodo.23903579","title":"A Structure for the Beal Conjecture","display_name":"A Structure for the Beal Conjecture","publication_year":2026,"publication_date":"2026-11-03","ids":{"openalex":"https://openalex.org/W7225144666","doi":"https://doi.org/10.5281/zenodo.23903579"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.23903579","doi":"https://doi.org/10.5281/zenodo.23903579","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23903579","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","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.23903579","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":0.9740999937057495,"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":0.9740999937057495,"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":0.023099999874830246,"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/T14500","display_name":"Smarandache Functions and Recreational Number Theory","score":0.0005000000237487257,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/beal-s-conjecture","display_name":"Beal's conjecture","score":0.8889999985694885},{"id":"https://openalex.org/keywords/fermat-s-last-theorem","display_name":"Fermat's Last Theorem","score":0.8569999933242798},{"id":"https://openalex.org/keywords/exponential-diophantine-equations","display_name":"exponential Diophantine equations","score":0.8949999809265137},{"id":"https://openalex.org/keywords/hyperbolic-geometry","display_name":"hyperbolic geometry","score":0.7329999804496765},{"id":"https://openalex.org/keywords/channels","display_name":"channels","score":0.4359999895095825},{"id":"https://openalex.org/keywords/number-of-phases","display_name":"number of phases","score":0.8479999899864197},{"id":"https://openalex.org/keywords/proof-by-contradiction","display_name":"reductio ad absurdum","score":0.6320000290870667}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7893999814987183},{"id":"https://openalex.org/C23230895","wikidata":"https://www.wikidata.org/wiki/Q104752","display_name":"Coprime integers","level":2,"score":0.741100013256073},{"id":"https://openalex.org/C2780388253","wikidata":"https://www.wikidata.org/wiki/Q5421508","display_name":"Exponent","level":2,"score":0.6439999938011169},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.5325999855995178},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.49799999594688416},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4666000008583069},{"id":"https://openalex.org/C169654258","wikidata":"https://www.wikidata.org/wiki/Q12479","display_name":"Number theory","level":2,"score":0.4259999990463257},{"id":"https://openalex.org/C2781204021","wikidata":"https://www.wikidata.org/wiki/Q6497091","display_name":"Lattice (music)","level":2,"score":0.37459999322891235},{"id":"https://openalex.org/C2780129039","wikidata":"https://www.wikidata.org/wiki/Q1931107","display_name":"Section (typography)","level":2,"score":0.3456000089645386},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.3303999900817871},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.3091999888420105},{"id":"https://openalex.org/C41608201","wikidata":"https://www.wikidata.org/wiki/Q980509","display_name":"Embedding","level":2,"score":0.3077000081539154},{"id":"https://openalex.org/C199879729","wikidata":"https://www.wikidata.org/wiki/Q205140","display_name":"Infinite set","level":3,"score":0.29739999771118164},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.27480000257492065},{"id":"https://openalex.org/C203492994","wikidata":"https://www.wikidata.org/wiki/Q909669","display_name":"Divisor (algebraic geometry)","level":2,"score":0.2639999985694885},{"id":"https://openalex.org/C171988527","wikidata":"https://www.wikidata.org/wiki/Q188276","display_name":"Pigeonhole principle","level":2,"score":0.26170000433921814},{"id":"https://openalex.org/C7321624","wikidata":"https://www.wikidata.org/wiki/Q205","display_name":"Infinity","level":2,"score":0.25929999351501465},{"id":"https://openalex.org/C136979486","wikidata":"https://www.wikidata.org/wiki/Q773483","display_name":"Existential quantification","level":2,"score":0.25060001015663147}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.23903579","doi":"https://doi.org/10.5281/zenodo.23903579","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23903579","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.23903580","doi":"https://doi.org/10.5281/zenodo.23903580","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23903580","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.23903579","doi":"https://doi.org/10.5281/zenodo.23903579","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23903579","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/W7155521213","https://openalex.org/W7223002102","https://openalex.org/W6949706502","https://openalex.org/W7172263724","https://openalex.org/W7105736771","https://openalex.org/W6941499442","https://openalex.org/W7137009074","https://openalex.org/W7131379186","https://openalex.org/W7110848303","https://openalex.org/W7114778979"],"abstract_inverted_index":{"This":[0],"manuscript":[1,239],"records":[2],"a":[3,17,20,37,53,60,88,108,113,134,152,199],"structural":[4],"program":[5],"for":[6,126,146,249],"the":[7,64,70,78,81,101,118,147,184,209,221,238,244],"Beal":[8],"conjecture":[9,245],"whose":[10],"guiding":[11],"principle":[12],"is":[13,16,52,186,225,229,247],"that":[14,133,157,211],"infinity":[15],"direction,":[18],"not":[19],"destination:":[21],"computation":[22],"can":[23],"verify":[24],"finite":[25],"windows,":[26],"while":[27],"any":[28],"statement":[29,228],"about":[30],"infinitely":[31],"many":[32],"cases":[33],"must":[34],"come":[35],"from":[36],"structure.":[38],"We":[39,74,116,129],"prove":[40,132],"an":[41,93,203,212],"exact":[42],"\\emph{seed":[43],"theorem}:":[44],"every":[45],"solution":[46],"of":[47,69,80,155,173,202],"$A^x+B^y=C^z$,":[48],"coprime":[49,54,190],"or":[50,236],"not,":[51],"primitive":[55],"triple":[56,83,205],"$u+v=w$":[57],"scaled":[58],"by":[59,214],"common":[61],"factor,":[62],"and":[63,106,124,150,182,206,246],"exponents":[65],"determine":[66],"which":[67,92],"terms":[68,160],"seed":[71],"are":[72],"constrained.":[73],"connect":[75],"this":[76],"to":[77,87,100,107,218],"geometry":[79],"exponent":[82,94,174,204],"(spherical,":[84],"flat,":[85],"hyperbolic),":[86],"``cube--cell''":[89],"model":[90],"in":[91],"below":[95],"$3$":[96],"collapses":[97],"one":[98,139],"dimension":[99],"lattice":[102,114],"unit":[103],"(a":[104],"\\emph{sheet}),":[105],"flat-case":[109],"descent":[110],"read":[111],"as":[112,193,231],"scaling.":[115],"develop":[117],"author's":[119],"\\emph{start}":[120],"approach:":[121],"fix":[122],"$A^x$":[123],"ask":[125],"all":[127,168],"hits.":[128],"define":[130],"\\emph{channels},":[131],"start":[135],"has":[136],"at":[137],"most":[138],"hit":[140],"per":[141],"channel,":[142],"derive":[143],"closed":[144],"forms":[145],"main":[148],"channels,":[149],"give":[151],"complete":[153],"classification":[154],"collisions":[156],"share":[158],"two":[159],"(the":[161,188],"\\emph{Chain":[162],"Lemma}),":[163],"together":[164],"with":[165],"data":[166],"on":[167,243],"other":[169],"collisions.":[170],"A":[171,195],"map":[172],"triples":[175,191],"shows":[176],"where":[177,183,223],"known":[178],"theorems":[179],"have":[180,217],"leverage":[181],"structure":[185],"silent":[187],"pairwise":[189],"such":[192],"$(3,4,5)$).":[194],"final":[196],"section":[197],"introduces":[198],"\\emph{phase":[200],"number}":[201],"sets":[207],"out":[208],"architecture":[210],"argument":[213],"contradiction":[215],"would":[216],"follow,":[219],"including":[220],"point":[222],"it":[224],"incomplete.":[226],"Every":[227],"labelled":[230],"known,":[232],"derived,":[233],"empirical,":[234],"heuristic":[235],"open;":[237],"asserts":[240],"no":[241],"verdict":[242],"offered":[248],"assessment.":[250]},"counts_by_year":[],"updated_date":"2026-11-05T12:25:35.769452","created_date":"2026-11-04T00:00:00"}
