{"id":"https://openalex.org/W7225807374","doi":"https://doi.org/10.5281/zenodo.24095211","title":"Multiplier obstructions for Legendre pairs of length 9p","display_name":"Multiplier obstructions for Legendre pairs of length 9p","publication_year":2026,"publication_date":"2026-11-11","ids":{"openalex":"https://openalex.org/W7225807374","doi":"https://doi.org/10.5281/zenodo.24095211"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.24095211","doi":"https://doi.org/10.5281/zenodo.24095211","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24095211","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":"Preprint"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.24095211","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5143659424","display_name":"Benedek Blackthorn","orcid":"https://orcid.org/0009-0004-3906-2919","observed_orcids":["https://orcid.org/0009-0004-3906-2919"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Blackthorn, Benedek","raw_affiliation_strings":["Independent researcher"],"raw_orcid":"https://orcid.org/0009-0004-3906-2919","affiliations":[{"raw_affiliation_string":"Independent researcher","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5143659424"],"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/T11130","display_name":"Coding theory and cryptography","score":0.9185000061988831,"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/T11130","display_name":"Coding theory and cryptography","score":0.9185000061988831,"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":0.044599998742341995,"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/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.012199999764561653,"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"}}],"keywords":[],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8537999987602234},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.697700023651123},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.5866000056266785},{"id":"https://openalex.org/C9991821","wikidata":"https://www.wikidata.org/wiki/Q858520","display_name":"Divisibility rule","level":2,"score":0.5401999950408936},{"id":"https://openalex.org/C124584101","wikidata":"https://www.wikidata.org/wiki/Q1053266","display_name":"Multiplier (economics)","level":2,"score":0.5023000240325928},{"id":"https://openalex.org/C111458787","wikidata":"https://www.wikidata.org/wiki/Q215405","display_name":"Legendre polynomials","level":2,"score":0.4496000111103058},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.41280001401901245},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.4027999937534332},{"id":"https://openalex.org/C60292330","wikidata":"https://www.wikidata.org/wiki/Q1014065","display_name":"Hadamard transform","level":2,"score":0.39500001072883606},{"id":"https://openalex.org/C34718186","wikidata":"https://www.wikidata.org/wiki/Q209875","display_name":"Binomial coefficient","level":2,"score":0.38499999046325684},{"id":"https://openalex.org/C61994974","wikidata":"https://www.wikidata.org/wiki/Q2297602","display_name":"Integer sequence","level":3,"score":0.3824000060558319},{"id":"https://openalex.org/C30463267","wikidata":"https://www.wikidata.org/wiki/Q1422682","display_name":"Hadamard matrix","level":3,"score":0.37790000438690186},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.36250001192092896},{"id":"https://openalex.org/C169654258","wikidata":"https://www.wikidata.org/wiki/Q12479","display_name":"Number theory","level":2,"score":0.3562999963760376},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.35510000586509705},{"id":"https://openalex.org/C2777759810","wikidata":"https://www.wikidata.org/wiki/Q149316","display_name":"Lemma (botany)","level":3,"score":0.33500000834465027},{"id":"https://openalex.org/C23230895","wikidata":"https://www.wikidata.org/wiki/Q104752","display_name":"Coprime integers","level":2,"score":0.3158000111579895},{"id":"https://openalex.org/C54732982","wikidata":"https://www.wikidata.org/wiki/Q1415345","display_name":"Modulo","level":2,"score":0.3059999942779541},{"id":"https://openalex.org/C2780757906","wikidata":"https://www.wikidata.org/wiki/Q5276676","display_name":"Dilation (metric space)","level":2,"score":0.28450000286102295},{"id":"https://openalex.org/C32795469","wikidata":"https://www.wikidata.org/wiki/Q5156430","display_name":"Complementary sequences","level":2,"score":0.27970001101493835},{"id":"https://openalex.org/C2781311116","wikidata":"https://www.wikidata.org/wiki/Q83306","display_name":"Group (periodic table)","level":2,"score":0.26930001378059387},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.26840001344680786},{"id":"https://openalex.org/C30860621","wikidata":"https://www.wikidata.org/wiki/Q49008","display_name":"Prime factor","level":3,"score":0.25839999318122864},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.25609999895095825}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.24095211","doi":"https://doi.org/10.5281/zenodo.24095211","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24095211","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":"Preprint"},{"id":"doi:10.5281/zenodo.24095212","doi":"https://doi.org/10.5281/zenodo.24095212","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24095212","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":"Preprint"}],"best_oa_location":{"id":"doi:10.5281/zenodo.24095211","doi":"https://doi.org/10.5281/zenodo.24095211","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24095211","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":"Preprint"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"A":[0,90,136,152],"Legendre":[1,123,210],"pair":[2,106,124,200,211],"of":[3,8,27,41,125,157,160,168,175,212,292,305,317,429,451],"odd":[4,54,68],"length":[5,50,108,126,161,213],"n":[6],"consists":[7],"two":[9],"\u00b11":[10],"sequences":[11,119,159,217],"whose":[12],"periodic":[13],"autocorrelations":[14],"sum":[15,134],"to":[16,78,107],"\u22122":[17],"at":[18,49,182,229],"every":[19],"nonzero":[20],"shift;":[21],"it":[22,66],"gives":[23,93,290],"a":[24,39,60,122,166,173,318,323,373],"Hadamard":[25],"matrix":[26],"order":[28,69],"2n":[29],"+":[30,88,100,115,186],"2.":[31],"Searches":[32],"for":[33,52,155,232,474],"such":[34,59],"pairs":[35,156],"often":[36],"assume":[37],"that":[38,48,140,189,197,208,277],"group":[40,61],"multipliers":[42,75],"fixes":[43],"both":[44,216],"sequences.":[45],"We":[46,205],"show":[47,188,207],"9p,":[51],"an":[53,226,242],"prime":[55],"p":[56,83,95,183],"\u2260":[57],"3,":[58],"must":[62],"be":[63],"small:":[64],"if":[65],"has":[67,215],"h":[70,240],"\u2265":[71,84,96],"19":[72],"and":[73,128,170,172,223,259,270,300,309,315,322,343,365,369,379,385,395,398,415,421,437,460,468],"its":[74,260,297,380],"are":[76,248,389,422],"congruent":[77],"1":[79,187],"modulo":[80],"9,":[81],"then":[82],"6h\u00b2":[85,185],"\u2212":[86,98],"4h":[87],"1.":[89,101],"first-moment":[91],"argument":[92],"only":[94,359],"h(h":[97],"1)/2":[99],"The":[102,264,326,355,377,425,470],"proof":[103,284,352],"compresses":[104],"the":[105,118,129,149,190,195,198,220,257,268,272,286,293,302,306,310,313,329,340,345,349,360,383,386,396,399,403,430,438,464],"nine.":[109],"Each":[110],"compressed":[111,199],"sequence":[112],"is":[113,194,472],"Z":[114,120],"2hY,":[116],"where":[117],"form":[121],"nine":[127],"integer":[130,158],"residuals":[131,142],"Y":[132],"have":[133,144],"zero.":[135],"variance":[137],"bound":[138,228],"shows":[139],"small":[141],"would":[143],"constant":[145],"combined":[146],"autocorrelation":[147],"off":[148],"zero":[150],"shift.":[151],"divisibility":[153],"theorem":[154,281],"nine,":[162],"which":[163,289,445],"follows":[164],"from":[165],"lemma":[167,174],"Arasu":[169],"Xiang":[171],"Ma,":[176],"rules":[177],"this":[178],"out.":[179],"Integer":[180],"solutions":[181],"=":[184],"leading":[191],"coefficient":[192],"six":[193],"best":[196],"equation":[201],"alone":[202],"can":[203],"give.":[204],"also":[206,249],"no":[209,283],"279":[214],"fixed":[218],"by":[219],"multiplier":[221],"64,":[222],"we":[224],"prove":[225],"analogous":[227],"lengths":[230],"\u2113\u00b2p":[231],"primes":[233],"\u2113":[234],"\u2261":[235],"3":[236,362],"(mod":[237],"4)":[238],"when":[239],"exceeds":[241],"explicit":[243],"threshold.":[244],"All":[245],"numbered":[246],"results":[247],"proved":[250],"in":[251,339],"Lean":[252,273,331,341,351,400],"4.":[253],"This":[254],"record":[255,314],"holds":[256],"article":[258,378],"source":[261,265,370],"archive,":[262],"supplement.zip.":[263],"archive":[266],"contains":[267],"manuscript":[269],"bibliography;":[271],"proofs,":[274,466],"with":[275,282,296,348,463],"files":[276,388],"state":[278],"each":[279,291,408],"final":[280],"inside;":[285],"claim":[287],"map,":[288],"eight":[294],"claims":[295],"hypotheses,":[298],"status":[299],"limits;":[301],"pinned":[303],"versions":[304],"formal":[307,320,375],"tools":[308],"replay":[311,327],"commands;":[312],"receipt":[316],"passing":[319],"replay;":[321],"check":[324,356],"script.":[325],"compiled":[328],"first-party":[330],"modules":[332],"afresh,":[333],"audited":[334],"their":[335,418,447],"axioms,":[336],"replayed":[337],"them":[338],"kernel":[342],"checked":[344],"exported":[346],"dependencies":[347],"independent":[350],"checker":[353],"nanoda.":[354],"script":[357],"uses":[358],"Python":[361],"standard":[363],"library":[364],"covers":[366],"finite":[367],"arithmetic":[368],"consistency,":[371],"not":[372,423],"fresh":[374],"replay.":[376],"LaTeX":[381],"source,":[382],"documentation":[384],"data":[387],"licensed":[390],"under":[391,402],"CC":[392],"BY":[393],"4.0,":[394],"programs":[397,467],"development":[401],"MIT":[404],"License;":[405],"LICENSES.md":[406],"assigns":[407],"file.":[409],"Cited":[410],"works,":[411],"Lean,":[412],"Mathlib,":[413],"lean4export":[414],"nanoda":[416],"keep":[417,446],"own":[419,448],"terms":[420],"included.":[424],"PDF":[426],"embeds":[427],"subsets":[428],"Latin":[431],"Modern":[432],"fonts":[433,440],"(GUST":[434],"Font":[435,443],"License)":[436],"AMS":[439],"(SIL":[441],"Open":[442],"License),":[444],"terms.":[449],"Use":[450],"generative":[452],"AI.":[453],"Generative":[454],"AI":[455],"tools,":[456],"including":[457],"ChatGPT,":[458],"Codex":[459],"Claude,":[461],"assisted":[462],"mathematics,":[465],"text.":[469],"author":[471],"responsible":[473],"all":[475],"content.":[476]},"counts_by_year":[],"updated_date":"2026-11-12T12:28:57.059296","created_date":"2026-11-12T00:00:00"}