{"id":"https://openalex.org/W7225610546","doi":"https://doi.org/10.33774/coe-2026-5615j","title":"Fermat's Last Theorem \u2026 (GN_+)","display_name":"Fermat's Last Theorem \u2026 (GN_+)","publication_year":2026,"publication_date":"2026-11-09","ids":{"openalex":"https://openalex.org/W7225610546","doi":"https://doi.org/10.33774/coe-2026-5615j"},"language":"en","primary_location":{"id":"doi:10.33774/coe-2026-5615j","doi":"https://doi.org/10.33774/coe-2026-5615j","is_oa":true,"landing_page_url":"https://doi.org/10.33774/coe-2026-5615j","pdf_url":"https://www.cambridge.org/engage/api-gateway/coe/assets/orp/resource/item/6ad3cdc1f4ef06beb657eb96/original/fermat-s-last-theorem-gn.pdf","source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"posted-content"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://www.cambridge.org/engage/api-gateway/coe/assets/orp/resource/item/6ad3cdc1f4ef06beb657eb96/original/fermat-s-last-theorem-gn.pdf","any_repository_has_fulltext":null},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5035299801","display_name":"Taha Muhammad","orcid":"https://orcid.org/0000-0002-8954-9264","observed_orcids":["https://orcid.org/0000-0002-8954-9264"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Taha Muhammad","raw_affiliation_strings":[],"raw_orcid":"https://orcid.org/0000-0002-8954-9264","affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5035299801"],"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":0.9291999936103821,"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":0.9291999936103821,"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":0.028699999675154686,"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/T12504","display_name":"Mathematics and Applications","score":0.024700000882148743,"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/fermat-s-last-theorem","display_name":"Fermat's Last Theorem","score":1.0},{"id":"https://openalex.org/keywords/positive-integers","display_name":"positive integers","score":0.5},{"id":"https://openalex.org/keywords/triangle-inequality","display_name":"triangle inequality","score":0.675000011920929},{"id":"https://openalex.org/keywords/right-triangles","display_name":"right triangles","score":0.8759999871253967},{"id":"https://openalex.org/keywords/acute-triangles","display_name":"acute triangles","score":0.9430000185966492},{"id":"https://openalex.org/keywords/obtuse-triangles","display_name":"obtuse triangles","score":1.0},{"id":"https://openalex.org/keywords/set-partitioning-problem","display_name":"set partitioning","score":0.5049999952316284}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7631000280380249},{"id":"https://openalex.org/C182964748","wikidata":"https://www.wikidata.org/wiki/Q208216","display_name":"Triangle inequality","level":2,"score":0.6243000030517578},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5601999759674072},{"id":"https://openalex.org/C42812","wikidata":"https://www.wikidata.org/wiki/Q1082910","display_name":"Partition (number theory)","level":2,"score":0.4991999864578247},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.42399999499320984},{"id":"https://openalex.org/C119050217","wikidata":"https://www.wikidata.org/wiki/Q17101806","display_name":"Intersection theorem","level":5,"score":0.4036000072956085},{"id":"https://openalex.org/C171636804","wikidata":"https://www.wikidata.org/wiki/Q1149458","display_name":"Compactness theorem","level":4,"score":0.36410000920295715},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.3537999987602234},{"id":"https://openalex.org/C48540410","wikidata":"https://www.wikidata.org/wiki/Q5221089","display_name":"Danskin's theorem","level":4,"score":0.33899998664855957},{"id":"https://openalex.org/C206880738","wikidata":"https://www.wikidata.org/wiki/Q431667","display_name":"Automated theorem proving","level":2,"score":0.3149000108242035},{"id":"https://openalex.org/C72319582","wikidata":"https://www.wikidata.org/wiki/Q584304","display_name":"Degenerate energy levels","level":2,"score":0.30660000443458557},{"id":"https://openalex.org/C194886279","wikidata":"https://www.wikidata.org/wiki/Q430271","display_name":"Fundamental theorem","level":3,"score":0.295199990272522},{"id":"https://openalex.org/C139588767","wikidata":"https://www.wikidata.org/wiki/Q17098096","display_name":"Mean value theorem (divided differences)","level":4,"score":0.2854999899864197},{"id":"https://openalex.org/C146147875","wikidata":"https://www.wikidata.org/wiki/Q1144897","display_name":"Brouwer fixed-point theorem","level":3,"score":0.2827000021934509},{"id":"https://openalex.org/C180375552","wikidata":"https://www.wikidata.org/wiki/Q2635326","display_name":"Structured program theorem","level":2,"score":0.2793999910354614},{"id":"https://openalex.org/C182124507","wikidata":"https://www.wikidata.org/wiki/Q166154","display_name":"Line segment","level":2,"score":0.26660001277923584}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.33774/coe-2026-5615j","doi":"https://doi.org/10.33774/coe-2026-5615j","is_oa":true,"landing_page_url":"https://doi.org/10.33774/coe-2026-5615j","pdf_url":"https://www.cambridge.org/engage/api-gateway/coe/assets/orp/resource/item/6ad3cdc1f4ef06beb657eb96/original/fermat-s-last-theorem-gn.pdf","source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"posted-content"}],"best_oa_location":{"id":"doi:10.33774/coe-2026-5615j","doi":"https://doi.org/10.33774/coe-2026-5615j","is_oa":true,"landing_page_url":"https://doi.org/10.33774/coe-2026-5615j","pdf_url":"https://www.cambridge.org/engage/api-gateway/coe/assets/orp/resource/item/6ad3cdc1f4ef06beb657eb96/original/fermat-s-last-theorem-gn.pdf","source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"posted-content"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":false},"content_urls":{"pdf":"https://content.openalex.org/works/W7225610546.pdf"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W4413329138","https://openalex.org/W4412495954","https://openalex.org/W4289529553","https://openalex.org/W4387264809","https://openalex.org/W7118218751","https://openalex.org/W2998327123","https://openalex.org/W2582040308","https://openalex.org/W6930663998","https://openalex.org/W2584639199","https://openalex.org/W6912640373"],"abstract_inverted_index":{"This":[0],"paper":[1],"introduces":[2],"a":[3],"geometric":[4,41],"and":[5,42,51,68],"set-theoretic":[6],"framework,":[7],"titled":[8],"\"Taha\u2019s":[9],"Geometry":[10],"Shapes":[11],"&":[12],"\\(\\mathbb{N}_{+}\\)":[13,37],"Fact,\"":[14],"to":[15],"evaluate":[16],"Fermat's":[17],"Last":[18],"Theorem":[19],"(\\(a^n":[20],"+":[21],"b^n":[22],"=":[23],"c^n\\)":[24],"for":[25],"\\(n":[26],">":[27],"2\\)).":[28],"We":[29],"partition":[30],"the":[31,47],"infinite":[32],"set":[33],"of":[34],"positive":[35],"integers":[36],"into":[38],"five":[39],"distinct":[40],"structural":[43],"subsets":[44],"based":[45],"on":[46],"triangle":[48],"inequality":[49],"theorem":[50],"segment":[52],"relations:":[53],"Right":[54],"Triangles":[55,58,61],"(\\(S_{1}\\)),":[56],"Acute":[57],"(\\(S_{2}\\)),":[59],"Obtuse":[60],"(\\(S_{3}\\)),":[62],"Perfect":[63],"Segments":[64,70],"where":[65,71],"\\(a+b=c\\)":[66],"(\\(S_{4}\\)),":[67],"Degenerate":[69],"\\(a+b":[72],"2\\).":[73]},"counts_by_year":[],"updated_date":"2026-11-11T12:16:26.678958","created_date":"2026-11-10T00:00:00"}