{"id":"https://openalex.org/W7225791640","doi":"https://doi.org/10.5281/zenodo.24095231","title":"Multiplier and reflection obstructions for Legendre pairs of length 9p with prescribed 9-compression","display_name":"Multiplier and reflection obstructions for Legendre pairs of length 9p with prescribed 9-compression","publication_year":2026,"publication_date":"2026-11-11","ids":{"openalex":"https://openalex.org/W7225791640","doi":"https://doi.org/10.5281/zenodo.24095231"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.24095231","doi":"https://doi.org/10.5281/zenodo.24095231","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24095231","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.24095231","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":0.5234000086784363,"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":0.5234000086784363,"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":0.384799987077713,"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":0.013700000010430813,"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":0.8693000078201294},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.625},{"id":"https://openalex.org/C165656649","wikidata":"https://www.wikidata.org/wiki/Q748339","display_name":"Legendre symbol","level":5,"score":0.4602999985218048},{"id":"https://openalex.org/C111458787","wikidata":"https://www.wikidata.org/wiki/Q215405","display_name":"Legendre polynomials","level":2,"score":0.45260000228881836},{"id":"https://openalex.org/C54732982","wikidata":"https://www.wikidata.org/wiki/Q1415345","display_name":"Modulo","level":2,"score":0.44940000772476196},{"id":"https://openalex.org/C135598885","wikidata":"https://www.wikidata.org/wiki/Q1366302","display_name":"Row","level":2,"score":0.4146000146865845},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.37549999356269836},{"id":"https://openalex.org/C87670640","wikidata":"https://www.wikidata.org/wiki/Q132469","display_name":"Fermat's Last Theorem","level":2,"score":0.37540000677108765},{"id":"https://openalex.org/C48372109","wikidata":"https://www.wikidata.org/wiki/Q3913","display_name":"Binary number","level":2,"score":0.375},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.3727000057697296},{"id":"https://openalex.org/C124584101","wikidata":"https://www.wikidata.org/wiki/Q1053266","display_name":"Multiplier (economics)","level":2,"score":0.3711000084877014},{"id":"https://openalex.org/C181633103","wikidata":"https://www.wikidata.org/wiki/Q1834342","display_name":"Modular group","level":2,"score":0.36739999055862427},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.3578999936580658},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3544999957084656},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.3472000062465668},{"id":"https://openalex.org/C65682993","wikidata":"https://www.wikidata.org/wiki/Q1056451","display_name":"Reflection (computer programming)","level":2,"score":0.32679998874664307},{"id":"https://openalex.org/C23230895","wikidata":"https://www.wikidata.org/wiki/Q104752","display_name":"Coprime integers","level":2,"score":0.3231000006198883},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.3215999901294708},{"id":"https://openalex.org/C2781311116","wikidata":"https://www.wikidata.org/wiki/Q83306","display_name":"Group (periodic table)","level":2,"score":0.30390000343322754},{"id":"https://openalex.org/C199479865","wikidata":"https://www.wikidata.org/wiki/Q205966","display_name":"Riemann hypothesis","level":2,"score":0.30379998683929443},{"id":"https://openalex.org/C12817185","wikidata":"https://www.wikidata.org/wiki/Q878259","display_name":"Quadratic residue","level":2,"score":0.3037000000476837},{"id":"https://openalex.org/C92757383","wikidata":"https://www.wikidata.org/wiki/Q382497","display_name":"Affine transformation","level":2,"score":0.3001999855041504},{"id":"https://openalex.org/C80695182","wikidata":"https://www.wikidata.org/wiki/Q864134","display_name":"Binary quadratic form","level":4,"score":0.29330000281333923},{"id":"https://openalex.org/C34718186","wikidata":"https://www.wikidata.org/wiki/Q209875","display_name":"Binomial coefficient","level":2,"score":0.28769999742507935},{"id":"https://openalex.org/C132074034","wikidata":"https://www.wikidata.org/wiki/Q154210","display_name":"Congruence (geometry)","level":2,"score":0.2831000089645386},{"id":"https://openalex.org/C92788228","wikidata":"https://www.wikidata.org/wiki/Q3777923","display_name":"Multiplicative group","level":3,"score":0.2531000077724457}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.24095231","doi":"https://doi.org/10.5281/zenodo.24095231","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24095231","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.24095232","doi":"https://doi.org/10.5281/zenodo.24095232","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24095232","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.24095231","doi":"https://doi.org/10.5281/zenodo.24095231","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24095231","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":[0,405],"binary":[1],"Legendre":[2,34,386],"pair":[3,35],"of":[4,9,36,65,90,121,144,147,163,166,191,204,211,288,356,380,388,438,499,585,671],"odd":[5,28],"length":[6,37,43,112,135,277,318,343,361,389,413,420,444,517,573,594],"n":[7],"consists":[8],"two":[10,40,196,239,365,576,638],"\u00b11":[11],"sequences":[12],"whose":[13,39,575],"periodic":[14],"autocorrelations":[15],"add":[16],"to":[17,42],"\u22122":[18],"at":[19,111,134,208,276,292,308,317,342,348,360,391,412,419,425,443,572,593],"every":[20,27],"nonzero":[21],"shift.":[22],"Kotsireas":[23],"conjectured":[24],"that":[25,77,97,110,133,169,289,430],"for":[26,177,225,231,385,516,590,694],"prime":[29],"p":[30,44,60,68,178,184,213,392,398],"there":[31],"is":[32,55,201,447,605,692],"a":[33,83,88,114,126,130,597,607],"9p":[38,216,390],"compressions":[41],"are":[45,366,625,648],"\u03b4":[46,50,62],"+":[47,217,223],"3\u03c7":[48],"and":[49,61,181,242,263,296,311,313,330,351,353,395,415,456,473,496,511,528,532,535,540,554,582,612,618,622,634,642,646,656,680,688],"\u2212":[51,221],"3\u03c7,":[52],"where":[53],"\u03c7":[54],"the":[56,63,94,104,142,151,161,167,172,189,192,195,202,205,212,247,260,265,271,274,279,286,298,305,314,340,345,354,357,369,381,408,416,428,436,439,454,464,467,478,481,484,497,500,507,512,519,549,556,568,583,586,616,623,637,643,651,657,661,684],"quadratic":[57],"character":[58],"modulo":[59],"indicator":[64],"zero.":[66],"For":[67,183],"\u2260":[69],"3":[70,186],"we":[71,108],"prove":[72],"restrictions":[73],"on":[74,93,171,402,407],"such":[75,234],"pairs":[76,387,411,418,432],"have":[78,117,252,322,578],"additional":[79],"structure.":[80],"We":[81],"give":[82],"field-descent":[84,272],"criterion":[85],"under":[86,427,627,650],"which":[87,229,446,503,521,537,596],"group":[89],"multipliers":[91,168],"acting":[92],"p-coordinate":[95,173,193],"alone":[96,174],"fixes":[98],"one":[99,199,210,434,543],"row":[100,115,153,200],"must":[101],"also":[102,606],"fix":[103],"other.":[105],"With":[106],"it":[107,137],"show":[109],"261":[113],"cannot":[116,138],"an":[118],"affine":[119],"symmetry":[120],"order":[122,145],"seven":[123],"and,":[124],"with":[125,156,256,282,301,325,371,469,477,487,683],"norm":[127,280],"equation":[128],"in":[129,368,548,660],"CM":[131],"field,":[132],"333":[136],"be":[139],"fixed":[140,159],"by":[141,160,259,264],"subgroup":[143,162],"nine":[146],"these":[148],"multipliers,":[149],"whatever":[150],"other":[152,206,619],"is.":[154],"Pairs":[155],"both":[157],"rows":[158],"index":[164],"six":[165],"act":[170],"exist":[175],"exactly":[176],"=":[179,219,393,399],"7":[180],"13.":[182],"\u2261":[185],"(mod":[187],"4)":[188],"reflection":[190,203],"exchanges":[194],"compressions.":[197],"If":[198],"outside":[207],"most":[209],"columns,":[214],"then":[215],"1":[218],"u\u00b2":[220],"uv":[222],"v\u00b2":[224],"integers":[226],"u,":[227],"v,":[228],"fails":[230],"almost":[232],"all":[233,562,695],"primes.":[235],"Further":[236],"results":[237,307],"treat":[238],"exceptional":[240],"columns":[241],"several":[243],"individual":[244],"lengths.":[245],"Of":[246],"55":[248],"numbered":[249],"results,":[250],"42":[251],"Lean":[253,261,485,603,644],"4":[254],"proofs":[255,324],"Mathlib,":[257],"checked":[258],"kernel":[262],"independent":[266],"checker":[267,635],"nanoda.":[268],"These":[269,338],"include":[270,339],"criterion,":[273],"exclusion":[275,355],"333,":[278],"condition":[281],"its":[283,302,396,423,457,505,526,613],"density":[284],"statement,":[285],"exclusions":[287,341],"two-column":[290,358],"case":[291,359,397],"lengths":[293,309,349],"63,":[294],"171":[295,350],"279,":[297],"dyadic":[299],"identity":[300],"four":[303],"families,":[304],"recognition":[306],"153":[310],"261,":[312,344,595],"rank-one":[315],"obstruction":[316],"117.":[319],"Another":[320],"11":[321],"written":[323],"exact":[326],"programs":[327,515,687],"or":[328],"searches":[329],"count":[331],"as":[332,667],"computer":[333,403],"evidence,":[334],"not":[335,448],"formal":[336],"proofs.":[337],"one-column":[346],"result":[347],"207,":[352],"207.":[362],"The":[363,374,460,546,602,610,631,690],"remaining":[364],"proved":[367],"text":[370],"external":[372],"dependencies.":[373],"index-six":[375],"theorem":[376],"uses":[377],"Theorem":[378],"1.2":[379],"article":[382,455,611],"Multiplier":[383],"obstructions":[384],"31,":[394],"19":[400],"rests":[401],"evidence.":[404],"proposition":[406],"released":[409,440],"control":[410],"63":[414],"prescribed":[417],"99":[421,426],"states":[422,504],"clause":[424],"hypothesis":[429],"those":[431,552],"form":[433],"orbit,":[435],"conclusion":[437],"exhaustive":[441],"search":[442,514,571],"99,":[445],"rerun":[449],"here.":[450],"This":[451],"record":[452],"holds":[453],"source":[458,461,527,544],"archive.":[459],"archive":[462],"contains":[463],"LaTeX":[465,614],"source;":[466],"data,":[468,636],"witnesses,":[470],"factor":[471],"certificates":[472],"finite":[474,564],"records":[475,647],"compared":[476],"replay":[479,482,490,494,524,600,604],"outputs;":[480],"programs;":[483],"sources,":[486],"their":[488,579,664],"pinned":[489],"profile,":[491],"component":[492],"map,":[493],"runner":[495],"receipt":[498],"passing":[501],"replay,":[502],"scope;":[506],"native":[508],"entry":[509],"points":[510],"three-word":[513,570],"153;":[518],"README,":[520],"gives":[522],"each":[523],"command,":[525],"data":[529,541,624],"locators,":[530],"controls":[531],"recorded":[533],"cost;":[534],"PUBLIC_MANIFEST.json,":[536],"pins":[538],"code":[539],"from":[542],"version.":[545],"command":[547],"README":[550,617],"authenticates":[551],"bytes":[553],"runs":[555,577],"routine":[557],"check.":[558],"That":[559],"check":[560],"reruns":[561],"declared":[563],"censuses":[565],"except":[566],"two:":[567],"compiled":[569],"153,":[574],"own":[580,665],"records,":[581],"traversal":[584],"empty":[587],"closed":[588],"balls":[589],"multiplier":[591],"233":[592],"separate":[598,608],"full":[599],"performs.":[601],"command.":[609],"source,":[615],"prose":[620],"documentation,":[621],"licensed":[626,649],"CC":[628],"BY":[629],"4.0.":[630],"programs,":[632],"settings":[633],"TeX":[639],"support":[640],"files,":[641],"sources":[645],"MIT":[652],"License.":[653],"Reused":[654],"material":[655],"fonts":[658],"embedded":[659],"PDF":[662],"keep":[663],"terms,":[666],"LICENSES.md":[668],"states.":[669],"Use":[670],"generative":[672],"AI.":[673],"Generative":[674],"AI":[675],"tools,":[676],"including":[677],"ChatGPT,":[678],"Codex":[679],"Claude,":[681],"assisted":[682],"mathematics,":[685],"proofs,":[686],"text.":[689],"author":[691],"responsible":[693],"content.":[696]},"counts_by_year":[],"updated_date":"2026-11-12T12:28:57.059296","created_date":"2026-11-12T00:00:00"}
