{"id":"https://openalex.org/W7203781147","doi":"https://doi.org/10.48550/arxiv.2608.24035","title":"Sieve dimension and search depth for the Erd\u0151s-Straus conjecture, $n \\equiv 1 \\pmod{24}$","display_name":"Sieve dimension and search depth for the Erd\u0151s-Straus conjecture, $n \\equiv 1 \\pmod{24}$","publication_year":2026,"publication_date":"2026-08-19","ids":{"openalex":"https://openalex.org/W7203781147","doi":"https://doi.org/10.48550/arxiv.2608.24035"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2608.24035","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.24035","pdf_url":"https://arxiv.org/pdf/2608.24035","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2608.24035","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5110458492","display_name":"Benjamin Dahan","orcid":"https://orcid.org/0009-0000-9332-3279","observed_orcids":["https://orcid.org/0009-0000-9332-3279"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Dahan, Benjamin","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5110458492"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":2,"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/T11166","display_name":"Analytic Number Theory Research","score":1.0,"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":1.0,"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.0,"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/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.0,"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/erdos-straus-conjecture","display_name":"Erd\u0151s\u2013Straus conjecture","score":0.9980000257492065},{"id":"https://openalex.org/keywords/search-depth","display_name":"search depth","score":0.8460000157356262},{"id":"https://openalex.org/keywords/covering-systems","display_name":"covering systems","score":0.453000009059906},{"id":"https://openalex.org/keywords/fixed-point-free-involutions","display_name":"fixed-point-free involutions","score":0.8769999742507935},{"id":"https://openalex.org/keywords/primitive-representations","display_name":"primitive representations","score":0.671999990940094}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8992000222206116},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6488000154495239},{"id":"https://openalex.org/C23230895","wikidata":"https://www.wikidata.org/wiki/Q104752","display_name":"Coprime integers","level":2,"score":0.6471999883651733},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.5232999920845032},{"id":"https://openalex.org/C2780388253","wikidata":"https://www.wikidata.org/wiki/Q5421508","display_name":"Exponent","level":2,"score":0.49050000309944153},{"id":"https://openalex.org/C203492994","wikidata":"https://www.wikidata.org/wiki/Q909669","display_name":"Divisor (algebraic geometry)","level":2,"score":0.46299999952316284},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.45680001378059387},{"id":"https://openalex.org/C78397625","wikidata":"https://www.wikidata.org/wiki/Q192487","display_name":"Discriminant","level":2,"score":0.4147999882698059},{"id":"https://openalex.org/C17732976","wikidata":"https://www.wikidata.org/wiki/Q7511782","display_name":"Sieve (category theory)","level":2,"score":0.4115999937057495},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.40779998898506165},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.39879998564720154},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.3882000148296356},{"id":"https://openalex.org/C128394911","wikidata":"https://www.wikidata.org/wiki/Q1987578","display_name":"Unit interval","level":2,"score":0.3684000074863434},{"id":"https://openalex.org/C2778067643","wikidata":"https://www.wikidata.org/wiki/Q166507","display_name":"Interval (graph theory)","level":2,"score":0.36399999260902405},{"id":"https://openalex.org/C30860621","wikidata":"https://www.wikidata.org/wiki/Q49008","display_name":"Prime factor","level":3,"score":0.33009999990463257},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.3239000141620636},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.3160000145435333},{"id":"https://openalex.org/C169654258","wikidata":"https://www.wikidata.org/wiki/Q12479","display_name":"Number theory","level":2,"score":0.29319998621940613},{"id":"https://openalex.org/C162838799","wikidata":"https://www.wikidata.org/wiki/Q596077","display_name":"Counterexample","level":2,"score":0.29089999198913574},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.2888999879360199},{"id":"https://openalex.org/C113429393","wikidata":"https://www.wikidata.org/wiki/Q49008","display_name":"Prime number","level":2,"score":0.28290000557899475},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.25999999046325684},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.25999999046325684},{"id":"https://openalex.org/C122637931","wikidata":"https://www.wikidata.org/wiki/Q118084","display_name":"Unit (ring theory)","level":2,"score":0.2590000033378601}],"mesh":[],"locations_count":4,"locations":[{"id":"pmh:oai:arXiv.org:2608.24035","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.24035","pdf_url":"https://arxiv.org/pdf/2608.24035","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2608.24035","doi":"https://doi.org/10.48550/arxiv.2608.24035","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2608.24035","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},{"id":"doi:10.5281/zenodo.22014413","doi":"https://doi.org/10.5281/zenodo.22014413","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.22014413","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.22014414","doi":"https://doi.org/10.5281/zenodo.22014414","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.22014414","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":"pmh:oai:arXiv.org:2608.24035","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.24035","pdf_url":"https://arxiv.org/pdf/2608.24035","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7203781147.pdf","grobid_xml":"https://content.openalex.org/works/W7203781147.grobid-xml"},"referenced_works_count":14,"referenced_works":["https://openalex.org/W1606209956","https://openalex.org/W2964545547","https://openalex.org/W4283318202","https://openalex.org/W4393967581","https://openalex.org/W4416242143","https://openalex.org/W4416681961","https://openalex.org/W6911018868","https://openalex.org/W7105540184","https://openalex.org/W7124130835","https://openalex.org/W7131394609","https://openalex.org/W7131668056","https://openalex.org/W7160493909","https://openalex.org/W7162303464","https://openalex.org/W7164138851"],"related_works":["https://openalex.org/W7204233692","https://openalex.org/W7213762086","https://openalex.org/W3171083572","https://openalex.org/W7166572208","https://openalex.org/W7157924842","https://openalex.org/W7215048624","https://openalex.org/W7204650968","https://openalex.org/W7202040484","https://openalex.org/W7168053797","https://openalex.org/W7214169903"],"abstract_inverted_index":{"For":[0],"primes":[1,30,132,203],"$n\\equiv1\\pmod{24}$":[2],"we":[3,118],"study":[4],"how":[5],"deep":[6],"an":[7,93,120,205,238],"explicit,":[8],"factorization-free":[9,196],"search":[10],"for":[11,25,61,201],"a":[12,108,184,190,195],"decomposition":[13],"of":[14,28,37,46,50,75,140,147,153,169,182,204,231,234,244],"$4/n$":[15],"into":[16],"three":[17],"unit":[18],"fractions":[19],"has":[20,102,156],"to":[21,157],"go.":[22],"Write":[23],"$E_2(N;J)$":[24],"the":[26,44,47,69,72,76,80,86,90,124,141,150,162,166,179,199,224,229,232,241,245],"set":[27],"such":[29],"$n\\le":[31,133],"N$":[32],"at":[33,39,114,136],"which":[34,79,137,222],"no":[35],"witness":[36],"depth":[38],"most":[40],"$J$":[41,64],"exists,":[42],"in":[43,228],"sense":[45],"two-parameter":[48],"criterion":[49,143],"Theorem":[51],"3.9":[52],"with":[53],"coprime":[54],"parameters":[55],"$u,a\\le":[56],"J$.":[57],"We":[58],"prove":[59],"that":[60,95,161],"every":[62,115],"fixed":[63,185],"$$|E_2(N;J)|":[65],"\\ll_J":[66],"\\frac{N}{(\\log":[67],"N)^{1+\\mathfrak{A}(J)/2}},$$":[68],"exponent":[70,103],"being":[71],"exact":[73],"dimension":[74],"covering":[77],"on":[78],"proof":[81,84],"rests.":[82],"The":[83,145,250],"replaces":[85],"subgroup":[87],"generated":[88],"by":[89,107,149],"prime":[91],"factors,":[92],"approach":[94],"breaks":[96],"down":[97],"as":[98,100],"soon":[99],"$(\\mathbb{Z}/4m)^{\\times}$":[101],"greater":[104],"than":[105,178],"$2$,":[106],"fixed-point-free":[109],"involution,":[110],"and":[111,174,236],"is":[112,165],"unconditional":[113,121],"$J$.":[116],"Second,":[117],"exhibit":[119],"obstruction.":[122],"At":[123],"shift":[125,186],"$c=7$":[126],"there":[127],"are":[128],"$\\asymp":[129],"N(\\log":[130],"N)^{-3/2}$":[131],"N$,":[134],"$n\\equiv1\\pmod{24}$,":[135],"both":[138],"branches":[139],"divisor":[142],"fail.":[144],"representation":[146],"$K_7=(n+7)/4$":[148],"principal":[151],"form":[152],"discriminant":[154],"$-7$":[155],"be":[158],"primitive,":[159],"so":[160],"relevant":[163],"input":[164],"primitive-representation":[167],"theorem":[168],"Fuchs,":[170],"Hsu,":[171],"Rickards,":[172],"Schindler":[173],"Stange":[175],"[25]":[176],"rather":[177],"classical":[180],"results":[181],"Iwaniec;":[183],"therefore":[187],"cannot":[188],"leave":[189],"finite":[191],"residual":[192],"set.":[193],"Third,":[194],"procedure":[197],"decides":[198],"conjecture":[200,251],"all":[202],"interval":[206],"$[N,2N]$.":[207],"Its":[208],"Type":[209,217,246],"II":[210,248],"pass":[211,219],"costs":[212,220],"$\\mathcal{O}(N(\\log":[213],"N)^{3})$":[214],"while":[215],"its":[216],"I":[218],"$\u0398(N^{2})$,":[221],"locates":[223],"whole":[225],"quadratic":[226],"cost":[227],"extraction":[230],"divisors":[233],"$4u^{2}d+1$":[235],"exhibits":[237],"asymmetry":[239],"between":[240],"two":[242],"halves":[243],"I/Type":[247],"dichotomy.":[249],"itself":[252],"remains":[253],"open.":[254]},"counts_by_year":[{"year":2026,"cited_by_count":2}],"updated_date":"2026-11-08T12:12:28.789561","created_date":"2026-08-20T00:00:00"}