{"id":"https://openalex.org/W1964086337","doi":"https://doi.org/10.1090/s0025-5718-97-00818-1","title":"Decomposing 40 billion integers by four tetrahedral numbers","display_name":"Decomposing 40 billion integers by four tetrahedral numbers","publication_year":1997,"publication_date":"1997-01-01","ids":{"openalex":"https://openalex.org/W1964086337","doi":"https://doi.org/10.1090/s0025-5718-97-00818-1","mag":"1964086337"},"language":"en","primary_location":{"id":"doi:10.1090/s0025-5718-97-00818-1","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-97-00818-1","pdf_url":"https://www.ams.org/mcom/1997-66-218/S0025-5718-97-00818-1/S0025-5718-97-00818-1.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://www.ams.org/mcom/1997-66-218/S0025-5718-97-00818-1/S0025-5718-97-00818-1.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5079808538","display_name":"Chung-Chiang Chou","orcid":null},"institutions":[{"id":"https://openalex.org/I70522481","display_name":"National Changhua University of Education","ror":"https://ror.org/005gkfa10","country_code":"TW","type":"education","lineage":["https://openalex.org/I70522481"]}],"countries":["TW"],"is_corresponding":false,"raw_author_name":"Chung-Chiang Chou","raw_affiliation_strings":["Department of Mathematics, National ChangHua University of Education, ChangHua 50058, Taiwan","Department of Mathematics, National Changhua University of Education, Changhua, 50058, Taiwan#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, National ChangHua University of Education, ChangHua 50058, Taiwan","institution_ids":["https://openalex.org/I70522481"]},{"raw_affiliation_string":"Department of Mathematics, National Changhua University of Education, Changhua, 50058, Taiwan#TAB#","institution_ids":["https://openalex.org/I70522481"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5047165687","display_name":"Yuefan Deng","orcid":"https://orcid.org/0000-0002-5224-3958"},"institutions":[{"id":"https://openalex.org/I1327163397","display_name":"State University of New York","ror":"https://ror.org/01q1z8k08","country_code":"US","type":"education","lineage":["https://openalex.org/I1327163397"]},{"id":"https://openalex.org/I59553526","display_name":"Stony Brook University","ror":"https://ror.org/05qghxh33","country_code":"US","type":"education","lineage":["https://openalex.org/I59553526"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Yuefan Deng","raw_affiliation_strings":["Center for Scientific Computing, State University of New York at Stony Brook, Stony Brook, New York 11794"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Center for Scientific Computing, State University of New York at Stony Brook, Stony Brook, New York 11794","institution_ids":["https://openalex.org/I1327163397","https://openalex.org/I59553526"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":3,"citation_normalized_percentile":{"value":0.08607595,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"66","issue":"218","first_page":"893","last_page":"901"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9979000091552734,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9979000091552734,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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":0.9972000122070312,"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/T11130","display_name":"Coding theory and cryptography","score":0.9926999807357788,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.6600732803344727},{"id":"https://openalex.org/keywords/annotation","display_name":"Annotation","score":0.5504570603370667},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.49650126695632935},{"id":"https://openalex.org/keywords/type","display_name":"Type (biology)","score":0.4277060031890869},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.3693797290325165},{"id":"https://openalex.org/keywords/database","display_name":"Database","score":0.36381882429122925},{"id":"https://openalex.org/keywords/biology","display_name":"Biology","score":0.08334749937057495}],"concepts":[{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.6600732803344727},{"id":"https://openalex.org/C2776321320","wikidata":"https://www.wikidata.org/wiki/Q857525","display_name":"Annotation","level":2,"score":0.5504570603370667},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.49650126695632935},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.4277060031890869},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.3693797290325165},{"id":"https://openalex.org/C77088390","wikidata":"https://www.wikidata.org/wiki/Q8513","display_name":"Database","level":1,"score":0.36381882429122925},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.08334749937057495},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1090/s0025-5718-97-00818-1","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-97-00818-1","pdf_url":"https://www.ams.org/mcom/1997-66-218/S0025-5718-97-00818-1/S0025-5718-97-00818-1.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.140.7839","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.140.7839","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.ams.sunysb.edu/~deng/paper%20with%20CCC%20on%20waring%20problems%20S0025-5718-97-00818-1.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.154.1065","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.154.1065","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.ams.org/mcom/1997-66-218/S0025-5718-97-00818-1/S0025-5718-97-00818-1.pdf","raw_type":"text"}],"best_oa_location":{"id":"doi:10.1090/s0025-5718-97-00818-1","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-97-00818-1","pdf_url":"https://www.ams.org/mcom/1997-66-218/S0025-5718-97-00818-1/S0025-5718-97-00818-1.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1964086337.pdf","grobid_xml":"https://content.openalex.org/works/W1964086337.grobid-xml"},"referenced_works_count":4,"referenced_works":["https://openalex.org/W1980358154","https://openalex.org/W1993562577","https://openalex.org/W2364481961","https://openalex.org/W7053360101"],"related_works":["https://openalex.org/W2361861616","https://openalex.org/W2263699433","https://openalex.org/W2377979023","https://openalex.org/W2218034408","https://openalex.org/W2392921965","https://openalex.org/W2358755282","https://openalex.org/W2625833328","https://openalex.org/W1533177136","https://openalex.org/W4380994516","https://openalex.org/W2556260348"],"abstract_inverted_index":{"Based":[0],"upon":[1],"a":[2,7,82,94,99,120,136],"computer":[3],"search":[4],"performed":[5],"on":[6],"massively":[8],"parallel":[9],"supercomputer,":[10],"we":[11],"found":[12],"that":[13,181],"any":[14],"integer":[15],"<inline-formula":[16,30,44,60,106,156,185],"content-type=\"math/mathml\">":[17,31,45,61,107,157,186],"<mml:math":[18,32,46,62,108,158,187],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[19,33,47,63,109,159,188],"alttext=\"n\">":[20],"<mml:semantics>":[21,35,49,67,111,165,201],"<mml:mi>n</mml:mi>":[22,170,207,222],"<mml:annotation":[23,37,51,73,113,174,226],"encoding=\"application/x-tex\">n</mml:annotation>":[24],"</mml:semantics>":[25,39,53,75,115,176,229],"</mml:math>":[26,40,54,76,116,177,230],"</inline-formula>":[27,41,55,77,117,178,231],"less":[28],"than":[29,59],"alttext=\"40\">":[34,48],"<mml:mn>40</mml:mn>":[36,50],"encoding=\"application/x-tex\">40</mml:annotation>":[38,52],"billion":[42],"(":[43],"B)":[56],"but":[57],"greater":[58],"alttext=\"343":[64],"comma":[65],"867\">":[66],"<mml:mrow>":[68,166,202],"<mml:mn>343</mml:mn>":[69],"<mml:mo>,</mml:mo>":[70],"<mml:mn>867</mml:mn>":[71],"</mml:mrow>":[72,173,214,216,225],"encoding=\"application/x-tex\">343,867</mml:annotation>":[74],"can":[78],"be":[79],"written":[80],"as":[81],"sum":[83],"of":[84,139,151,182],"four":[85],"or":[86],"fewer":[87],"tetrahedral":[88],"numbers.":[89],"This":[90],"result":[91,138],"has":[92],"established":[93],"new":[95,153],"upper":[96],"bound":[97],"for":[98,130],"conjecture":[100],"compared":[101,179],"to":[102],"an":[103],"older":[104],"one,":[105],"alttext=\"1\">":[110],"<mml:mn>1</mml:mn>":[112],"encoding=\"application/x-tex\">1</mml:annotation>":[114],"B,":[118],"obtained":[119],"year":[121],"earlier.":[122],"It":[123],"also":[124],"gives":[125],"more":[126],"accurate":[127],"asymptotic":[128],"forms":[129],"partitioning.":[131],"All":[132],"this":[133],"improvement":[134],"is":[135,155],"direct":[137],"algorithmic":[140],"advances":[141],"in":[142],"efficient":[143],"memory":[144],"and":[145],"cpu":[146],"utilizations.":[147],"The":[148],"heuristic":[149],"complexity":[150],"the":[152,183],"algorithm":[154],"alttext=\"upper":[160,189],"O":[161,190],"left-parenthesis":[162,191],"n":[163,192,199],"right-parenthesis\">":[164,200],"<mml:mi>O</mml:mi>":[167,203],"<mml:mo":[168,171,204,223],"stretchy=\"false\">(</mml:mo>":[169,205],"stretchy=\"false\">)</mml:mo>":[172,224],"encoding=\"application/x-tex\">O(n)</mml:annotation>":[175],"with":[180],"old,":[184],"Superscript":[193],"5":[194],"slash":[195],"3":[196],"Baseline":[197],"log":[198],"<mml:msup>":[206],"<mml:mrow":[208,211],"class=\"MJX-TeXAtom-ORD\">":[209,212],"<mml:mn>5</mml:mn>":[210],"<mml:mo>/</mml:mo>":[213],"<mml:mn>3</mml:mn>":[215],"</mml:msup>":[217],"<mml:mi>log</mml:mi>":[218],"<mml:mo>":[219],"\u2061":[220],"</mml:mo>":[221],"encoding=\"application/x-tex\">O(n^{5/3}\\log":[227],"n)</mml:annotation>":[228],".":[232]},"counts_by_year":[{"year":2024,"cited_by_count":1}],"updated_date":"2026-07-22T07:51:19.307946","created_date":"2025-10-10T00:00:00"}