{"id":"https://openalex.org/W7225005768","doi":"https://doi.org/10.5281/zenodo.23790516","title":"On sums of four squares with |x+3y\u22125z| a power of 4","display_name":"On sums of four squares with |x+3y\u22125z| a power of 4","publication_year":2026,"publication_date":"2026-10-30","ids":{"openalex":"https://openalex.org/W7225005768","doi":"https://doi.org/10.5281/zenodo.23790516"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.23790516","doi":"https://doi.org/10.5281/zenodo.23790516","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23790516","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":"article-journal"},"volume":null,"type":"article","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.23790516","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5160256880","display_name":"Aaron Cooper","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Cooper, Aaron","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5160256880"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.93341855,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"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":0.9423999786376953,"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":0.9423999786376953,"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/T11428","display_name":"Advanced Mathematical Identities","score":0.030899999663233757,"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":0.020999999716877937,"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/sums-of-four-squares","display_name":"sums of four squares","score":0.8960000276565552},{"id":"https://openalex.org/keywords/sun-s-conjecture","display_name":"Sun's conjecture","score":0.6679999828338623},{"id":"https://openalex.org/keywords/quadratic-forms","display_name":"quadratic forms","score":0.4909999966621399},{"id":"https://openalex.org/keywords/imaginary-quadratic-fields","display_name":"imaginary quadratic fields","score":0.7009999752044678},{"id":"https://openalex.org/keywords/generalized-riemann-hypothesis","display_name":"generalized Riemann hypothesis","score":0.8339999914169312},{"id":"https://openalex.org/keywords/quadratic-twists","display_name":"quadratic twists","score":0.5709999799728394},{"id":"https://openalex.org/keywords/computer-assisted-proof","display_name":"computer-assisted proof","score":0.7329999804496765}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.853600025177002},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.6638000011444092},{"id":"https://openalex.org/C131220774","wikidata":"https://www.wikidata.org/wiki/Q1441456","display_name":"Ramanujan's sum","level":2,"score":0.5048999786376953},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.4909999966621399},{"id":"https://openalex.org/C199479865","wikidata":"https://www.wikidata.org/wiki/Q205966","display_name":"Riemann hypothesis","level":2,"score":0.4758000075817108},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.46869999170303345},{"id":"https://openalex.org/C2778355321","wikidata":"https://www.wikidata.org/wiki/Q17079427","display_name":"Identity (music)","level":2,"score":0.4375},{"id":"https://openalex.org/C95136341","wikidata":"https://www.wikidata.org/wiki/Q625519","display_name":"Quadratic field","level":4,"score":0.4244999885559082},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.41850000619888306},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.41339999437332153},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.38940000534057617},{"id":"https://openalex.org/C105546189","wikidata":"https://www.wikidata.org/wiki/Q1154787","display_name":"Theta function","level":2,"score":0.3725000023841858},{"id":"https://openalex.org/C169654258","wikidata":"https://www.wikidata.org/wiki/Q12479","display_name":"Number theory","level":2,"score":0.3578999936580658},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.3497999906539917},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.32989999651908875},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.3273000121116638},{"id":"https://openalex.org/C179603306","wikidata":"https://www.wikidata.org/wiki/Q268493","display_name":"Elliptic curve","level":2,"score":0.32199999690055847},{"id":"https://openalex.org/C80146680","wikidata":"https://www.wikidata.org/wiki/Q2654923","display_name":"Complex multiplication","level":3,"score":0.32190001010894775},{"id":"https://openalex.org/C46221946","wikidata":"https://www.wikidata.org/wiki/Q15051202","display_name":"Sums of powers","level":2,"score":0.3098999857902527},{"id":"https://openalex.org/C80695182","wikidata":"https://www.wikidata.org/wiki/Q864134","display_name":"Binary quadratic form","level":4,"score":0.295199990272522},{"id":"https://openalex.org/C163258240","wikidata":"https://www.wikidata.org/wiki/Q25342","display_name":"Power (physics)","level":2,"score":0.2946000099182129},{"id":"https://openalex.org/C95611797","wikidata":"https://www.wikidata.org/wiki/Q17502105","display_name":"Infimum and supremum","level":2,"score":0.2919999957084656},{"id":"https://openalex.org/C2776760102","wikidata":"https://www.wikidata.org/wiki/Q5139990","display_name":"Code (set theory)","level":3,"score":0.2727000117301941},{"id":"https://openalex.org/C184264201","wikidata":"https://www.wikidata.org/wiki/Q21199","display_name":"Natural number","level":2,"score":0.2660999894142151},{"id":"https://openalex.org/C7720571","wikidata":"https://www.wikidata.org/wiki/Q1136880","display_name":"Power of two","level":2,"score":0.26440000534057617}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.5281/zenodo.23790516","doi":"https://doi.org/10.5281/zenodo.23790516","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23790516","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":"article-journal"},{"id":"doi:10.5281/zenodo.23790517","doi":"https://doi.org/10.5281/zenodo.23790517","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23790517","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":"article-journal"},{"id":"doi:10.5281/zenodo.23851955","doi":"https://doi.org/10.5281/zenodo.23851955","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23851955","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":"article-journal"}],"best_oa_location":{"id":"doi:10.5281/zenodo.23790516","doi":"https://doi.org/10.5281/zenodo.23790516","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23790516","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":"article-journal"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W3091289437","https://openalex.org/W3119569741","https://openalex.org/W4416952536","https://openalex.org/W7212351980","https://openalex.org/W4388073111","https://openalex.org/W3092918472","https://openalex.org/W2394108147","https://openalex.org/W2962998647","https://openalex.org/W3005399550","https://openalex.org/W2901315739"],"abstract_inverted_index":{"Z.-W.":[0],"Sun":[1,96],"conjectured":[2],"that":[3,36,202],"every":[4],"positive":[5],"integer":[6],"n":[7,31,41,84,103,125,147,191,205],"can":[8,81],"be":[9],"written":[10],"as":[11],"x\u00b2+y\u00b2+z\u00b2+w\u00b2":[12],"with":[13,72,126],"x,":[14],"y,":[15],"z,":[16],"w":[17],"natural":[18],"numbers":[19],"and":[20,34,95,121,152,159,167,185,200,226,250,254],"|x+3y\u22125z|":[21],"a":[22,176,211],"power":[23],"of":[24,93,114,157,160,164,180,183,190,220,264],"4.":[25],"We":[26,173],"verify":[27],"the":[28,48,61,90,99,105,111,133,137,143,154,203,217,231,237,241,248,262,265,273],"conjecture":[29,100,144],"for":[30,39,101,122,145,150,153],"\u2264":[32],"2\u00b710\u00b9\u2070":[33],"show":[35,201],"it":[37,53],"holds":[38],"all":[40,102,123,146,251],"whose":[42],"16-free":[43],"part":[44,113],"is":[45,50,118,171,194,210],"sufficiently":[46],"large;":[47],"bound":[49],"ineffective,":[51],"but":[52,68,196],"becomes":[54],"effective":[55],"(provided":[56],"certain":[57],"implied":[58],"constants":[59],"in":[60,267],"literature":[62],"are":[63],"effective,":[64],"which":[65,80],"we":[66,97,141],"expect":[67],"have":[69],"not":[70,119,197],"checked)":[71],"at":[73,127],"most":[74,128],"one":[75],"exceptional":[76],"imaginary":[77],"quadratic":[78,155],"field,":[79],"only":[82],"affect":[83],"\u2261":[85,206],"5":[86,207],"(mod":[87,208],"8).":[88],"For":[89,136],"integral":[91],"version":[92,140],"Wu":[94],"prove":[98,142],"under":[104,132,148,236],"Generalized":[106],"Riemann":[107],"Hypothesis,":[108],"unconditionally":[109],"when":[110],"odd":[112],"70n\u22122":[115],"(resp.":[116],"70n\u221232)":[117],"squarefree,":[120],"large":[124],"two":[129],"exceptions":[130],"(effectively,":[131],"same":[134],"proviso).":[135],"original":[138],"natural-number":[139],"GRH":[149],"L(s,\u03c7_D)":[151],"twists":[156],"E\u2087\u2080":[158],"fifteen":[161],"explicit":[162],"newforms":[163],"weights":[165],"6":[166],"10;":[168],"this":[169],"proof":[170,179],"computer-assisted.":[172],"also":[174],"give":[175],"short":[177],"quaternion":[178],"an":[181],"identity":[182],"Li\u2013Zhou":[184],"Li\u2013Wang,":[186],"find":[187],"infinite":[188],"families":[189],"where":[192],"x+3y+5z=1":[193],"locally":[195],"globally":[198],"solvable,":[199],"class":[204],"8)":[209],"Ramanujan":[212],"x\u00b2+y\u00b2+10z\u00b2-type":[213],"problem":[214],"attached":[215],"to":[216,278],"elliptic":[218],"curve":[219],"conductor":[221],"70.":[222],"The":[223],"mathematics,":[224],"code":[225],"audits":[227],"were":[228],"produced":[229],"by":[230],"AI":[232],"system":[233],"Claude":[234],"(Anthropic)":[235],"author's":[238],"direction;":[239],"see":[240],"paper's":[242],"final":[243],"section.":[244],"This":[245],"record":[246],"contains":[247],"paper":[249],"programs,":[252],"certificates":[253],"logs.":[255],"Version":[256],"2":[257],"(1":[258],"Nov":[259],"2026):":[260],"corrects":[261],"argument":[263],"L-values":[266],"Sec.":[268],"7":[269],"(centre":[270],"l+1),":[271],"adds":[272],"hypothesis":[274],"v":[275],"\u2265":[276],"1":[277],"Lemma":[279],"2.2,":[280],"rewords":[281],"three":[282],"statements":[283],"about":[284],"computations;":[285],"no":[286],"theorem":[287],"changes.":[288]},"counts_by_year":[],"updated_date":"2026-11-02T09:45:03.842371","created_date":"2026-11-01T00:00:00"}