{"id":"https://openalex.org/W7214810486","doi":null,"title":"Solution of the Generalized Pythagorean Equation; x2+y2=n! and Formula ? by Number Theory","display_name":"Solution of the Generalized Pythagorean Equation; x2+y2=n! and Formula ? by Number Theory","publication_year":2026,"publication_date":"2026-06-24","ids":{"openalex":"https://openalex.org/W7214810486"},"language":"en","primary_location":{"id":"pmh:oai:ojs.asrjetsjournal.org:article/12240","doi":null,"is_oa":true,"landing_page_url":"https://asrjetsjournal.org/American_Scientific_Journal/article/view/12240","pdf_url":"https://asrjetsjournal.org/American_Scientific_Journal/article/download/12240/2944","source":{"id":"https://openalex.org/S7407065841","display_name":"American Scientific Research Journal for Engineering, Technology, and Sciences","issn_l":"2313-4402","issn":["2313-4402","2313-4410"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"2313-4410","raw_type":"info:eu-repo/semantics/article"},"volume":null,"type":"article","indexed_in":[],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://asrjetsjournal.org/American_Scientific_Journal/article/download/12240/2944","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5153512230","display_name":"Deval Sidi Mohamed","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I90527594","display_name":"Universit\u00e9 de Nouakchott","ror":"https://ror.org/00pt67k84","country_code":"MR","type":"education","lineage":["https://openalex.org/I90527594"]}],"countries":["MR"],"is_corresponding":true,"raw_author_name":"Deval Sidi Mohamed","raw_affiliation_strings":["Department of Mathematics , Faculty of Science and Technology , University of Nouakchott"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , Faculty of Science and Technology , University of Nouakchott","institution_ids":["https://openalex.org/I90527594"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5153512230"],"corresponding_institution_ids":["https://openalex.org/I90527594"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.82860795,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"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/T11166","display_name":"Analytic Number Theory Research","score":0.9937000274658203,"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"}},"topics":[{"id":"https://openalex.org/T11166","display_name":"Analytic Number Theory Research","score":0.9937000274658203,"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/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.00430000014603138,"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/T14500","display_name":"Smarandache Functions and Recreational Number Theory","score":0.00039999998989515007,"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/sum-of-divisors-function","display_name":"sum of divisors function","score":0.4830000102519989}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8465999960899353},{"id":"https://openalex.org/C80918210","wikidata":"https://www.wikidata.org/wiki/Q208225","display_name":"Pythagorean triple","level":3,"score":0.7422000169754028},{"id":"https://openalex.org/C100461760","wikidata":"https://www.wikidata.org/wiki/Q11518","display_name":"Pythagorean theorem","level":2,"score":0.7340999841690063},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.45730000734329224},{"id":"https://openalex.org/C12989436","wikidata":"https://www.wikidata.org/wiki/Q3291120","display_name":"Class number formula","level":5,"score":0.4519999921321869},{"id":"https://openalex.org/C149381693","wikidata":"https://www.wikidata.org/wiki/Q10843274","display_name":"Analytic number theory","level":2,"score":0.4169999957084656},{"id":"https://openalex.org/C169654258","wikidata":"https://www.wikidata.org/wiki/Q12479","display_name":"Number theory","level":2,"score":0.41359999775886536},{"id":"https://openalex.org/C89507098","wikidata":"https://www.wikidata.org/wiki/Q2039117","display_name":"Pythagorean trigonometric identity","level":5,"score":0.4050000011920929},{"id":"https://openalex.org/C2780861071","wikidata":"https://www.wikidata.org/wiki/Q1062934","display_name":"Character (mathematics)","level":2,"score":0.4032000005245209},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3797000050544739},{"id":"https://openalex.org/C52690540","wikidata":"https://www.wikidata.org/wiki/Q613048","display_name":"Algebraic number theory","level":3,"score":0.33390000462532043},{"id":"https://openalex.org/C169214877","wikidata":"https://www.wikidata.org/wiki/Q981016","display_name":"Dirichlet distribution","level":3,"score":0.329800009727478},{"id":"https://openalex.org/C178635117","wikidata":"https://www.wikidata.org/wiki/Q747499","display_name":"RADIUS","level":2,"score":0.3147999942302704},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3116999864578247},{"id":"https://openalex.org/C2779193601","wikidata":"https://www.wikidata.org/wiki/Q20026918","display_name":"Mathematical theory","level":2,"score":0.3073999881744385},{"id":"https://openalex.org/C184264201","wikidata":"https://www.wikidata.org/wiki/Q21199","display_name":"Natural number","level":2,"score":0.3037000000476837},{"id":"https://openalex.org/C153046414","wikidata":"https://www.wikidata.org/wiki/Q12482","display_name":"Set theory","level":3,"score":0.30300000309944153},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.2782000005245209},{"id":"https://openalex.org/C173734053","wikidata":"https://www.wikidata.org/wiki/Q47577","display_name":"Fibonacci number","level":2,"score":0.27129998803138733},{"id":"https://openalex.org/C126875415","wikidata":"https://www.wikidata.org/wiki/Q203425","display_name":"Unit circle","level":2,"score":0.25949999690055847},{"id":"https://openalex.org/C94020503","wikidata":"https://www.wikidata.org/wiki/Q12916","display_name":"Real number","level":2,"score":0.25679999589920044}],"mesh":[],"locations_count":1,"locations":[{"id":"pmh:oai:ojs.asrjetsjournal.org:article/12240","doi":null,"is_oa":true,"landing_page_url":"https://asrjetsjournal.org/American_Scientific_Journal/article/view/12240","pdf_url":"https://asrjetsjournal.org/American_Scientific_Journal/article/download/12240/2944","source":{"id":"https://openalex.org/S7407065841","display_name":"American Scientific Research Journal for Engineering, Technology, and Sciences","issn_l":"2313-4402","issn":["2313-4402","2313-4410"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"2313-4410","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":{"id":"pmh:oai:ojs.asrjetsjournal.org:article/12240","doi":null,"is_oa":true,"landing_page_url":"https://asrjetsjournal.org/American_Scientific_Journal/article/view/12240","pdf_url":"https://asrjetsjournal.org/American_Scientific_Journal/article/download/12240/2944","source":{"id":"https://openalex.org/S7407065841","display_name":"American Scientific Research Journal for Engineering, Technology, and Sciences","issn_l":"2313-4402","issn":["2313-4402","2313-4410"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"2313-4410","raw_type":"info:eu-repo/semantics/article"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214810486.pdf","grobid_xml":"https://content.openalex.org/works/W7214810486.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W2347946398","https://openalex.org/W7169663063","https://openalex.org/W1562342563","https://openalex.org/W2113491088","https://openalex.org/W2462970495","https://openalex.org/W1988511521","https://openalex.org/W7220693814","https://openalex.org/W6929903162","https://openalex.org/W6980742727","https://openalex.org/W1978649188"],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"we":[3,51],"are":[4],"interested":[5],"in":[6,59],"studying":[7],"the":[8,20,54,60,70],"generalized":[9],"Pythagorean":[10],"equation:":[11],"$$x^2+y^2=n!\\quad":[12],"\\quad":[13,29,31,35,37,41,43],"n\\in":[14],"\\mathbb{N}\\quad":[15],"(x,y)":[16],"\\in":[17],"\\mathbb{Z}^2.$$":[18],"Furthermore,":[19],"non-principe":[21],"Dirichlet":[22],"character":[23],"module":[24],"4,":[25],"defined":[26],"as":[27],"follows:$$\\chi(d)=\\left\\{\\begin{array}{c}1":[28],"if":[30,36,42],"d\\equiv":[32,38],"1(mod":[33],"4)\\\\-1":[34],"3(mod":[39],"4)\\\\0":[40],"d":[44],"\\>is":[45],"\\>even\\end{array}\\right.$$":[46],"Using":[47],"Jacobi's":[48],"Two-Square":[49],"Theorem,":[50],"show":[52],"that":[53],"total":[55],"number":[56,75],"of":[57,62],"lattices":[58],"circle":[61],"radius":[63],"$r=\\sqrt{N}$":[64],"is":[65,77],"equal":[66],"$\\sum^N_{n=1}\\sum_{d|n}\\chi(d)$.\\\\":[67],"Finally,":[68],"Application:":[69],"formula":[71],"for":[72],"$\\pi$":[73],"by":[74],"theory":[76],"given":[78],"by$$\\pi=\\lim_{N\\rightarrow":[79],"+\\infty}\\frac{4}{N}\\bigg(\\sum^N_{n=1}\\sum_{d|n}\\chi(d)\\bigg)=4(1-\\frac{1}{3}+\\frac{1}{5}-\\frac{1}{7}\\cdot\\cdot\\cdot\\cdot).$$":[80]},"counts_by_year":[],"updated_date":"2026-11-04T10:54:05.112534","created_date":"2026-09-30T00:00:00"}