{"id":"https://openalex.org/W2964252023","doi":"https://doi.org/10.4169/000298910x496769","title":"The Diophantine Equation <i> x <sup>4</sup> </i> \u00b1 <i> y <sup>4</sup> </i> = <i> iz <sup>2</sup> </i> in Gaussian Integers","display_name":"The Diophantine Equation <i> x <sup>4</sup> </i> \u00b1 <i> y <sup>4</sup> </i> = <i> iz <sup>2</sup> </i> in Gaussian Integers","publication_year":2010,"publication_date":"2010-07-22","ids":{"openalex":"https://openalex.org/W2964252023","doi":"https://doi.org/10.4169/000298910x496769","mag":"2964252023"},"language":"en","primary_location":{"id":"doi:10.4169/000298910x496769","is_oa":false,"landing_page_url":"https://doi.org/10.4169/000298910x496769","pdf_url":null,"source":{"id":"https://openalex.org/S156280010","display_name":"American Mathematical Monthly","issn_l":"0002-9890","issn":["0002-9890","1930-0972"],"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547"],"host_organization_lineage_names":["Taylor & Francis"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The American Mathematical Monthly","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5081214859","display_name":"Filip Najman","orcid":"https://orcid.org/0000-0002-0994-0846"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Filip Najman","raw_affiliation_strings":[","],"affiliations":[{"raw_affiliation_string":",","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5081214859"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":9,"citation_normalized_percentile":{"value":0.31269072,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"117","issue":"7","first_page":"637","last_page":"641"},"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":0.9865000247955322,"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/T12069","display_name":"Commutative Algebra and Its Applications","score":0.9804999828338623,"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/diophantine-equation","display_name":"Diophantine equation","score":0.8195232152938843},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.451314240694046},{"id":"https://openalex.org/keywords/elliptic-curve","display_name":"Elliptic curve","score":0.4153895080089569},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.37458401918411255},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.24676749110221863},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.20803716778755188}],"concepts":[{"id":"https://openalex.org/C206530611","wikidata":"https://www.wikidata.org/wiki/Q905896","display_name":"Diophantine equation","level":2,"score":0.8195232152938843},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.451314240694046},{"id":"https://openalex.org/C179603306","wikidata":"https://www.wikidata.org/wiki/Q268493","display_name":"Elliptic curve","level":2,"score":0.4153895080089569},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.37458401918411255},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.24676749110221863},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.20803716778755188}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.4169/000298910x496769","is_oa":false,"landing_page_url":"https://doi.org/10.4169/000298910x496769","pdf_url":null,"source":{"id":"https://openalex.org/S156280010","display_name":"American Mathematical Monthly","issn_l":"0002-9890","issn":["0002-9890","1930-0972"],"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547"],"host_organization_lineage_names":["Taylor & Francis"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The American Mathematical Monthly","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.212.7520","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.212.7520","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://web.math.hr/~fnajman/amm.pdf","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":10,"referenced_works":["https://openalex.org/W198005540","https://openalex.org/W1524918965","https://openalex.org/W1543374356","https://openalex.org/W1976564246","https://openalex.org/W2014478557","https://openalex.org/W2097381721","https://openalex.org/W2333976070","https://openalex.org/W2797332554","https://openalex.org/W2798196899","https://openalex.org/W2954945477"],"related_works":["https://openalex.org/W2935759653","https://openalex.org/W3105167352","https://openalex.org/W54078636","https://openalex.org/W2954470139","https://openalex.org/W1501425562","https://openalex.org/W2902782467","https://openalex.org/W3084825885","https://openalex.org/W2298861036","https://openalex.org/W3148032049","https://openalex.org/W3162755923"],"abstract_inverted_index":{"In":[0],"this":[1],"note":[2],"we":[3,27],"find":[4],"all":[5],"the":[6,9,24,36],"solutions":[7,46],"of":[8,32],"Diophantine":[10],"equation":[11,37],"x4":[12,38],"\u00b1":[13,39],"y4":[14,40],"=":[15,41],"iz2":[16],"using":[17,23],"elliptic":[18],"curves":[19],"over":[20],"\u211a(i).":[21],"Also,":[22],"same":[25],"method":[26],"give":[28],"a":[29],"new":[30],"proof":[31],"Hilbert's":[33],"result":[34],"that":[35],"z2":[42],"has":[43],"only":[44],"trivial":[45],"in":[47],"Gaussian":[48],"integers.":[49]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2015,"cited_by_count":2}],"updated_date":"2026-01-22T23:29:09.771500","created_date":"2025-10-10T00:00:00"}