{"id":"https://openalex.org/W7223012397","doi":"https://doi.org/10.48550/arxiv.2610.25661","title":"The factorial separation conjecture","display_name":"The factorial separation conjecture","publication_year":2026,"publication_date":"2026-10-19","ids":{"openalex":"https://openalex.org/W7223012397","doi":"https://doi.org/10.48550/arxiv.2610.25661"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.25661","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.25661","pdf_url":"https://arxiv.org/pdf/2610.25661","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/2610.25661","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5103936668","display_name":"Valmir Krasniqi","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Krasniqi, Valmir","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5103936668"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"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/T11166","display_name":"Analytic Number Theory Research","score":0.9994999766349792,"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":0.9994999766349792,"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":0.0005000000237487257,"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"}},{"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"}}],"keywords":[{"id":"https://openalex.org/keywords/factorials","display_name":"factorials","score":0.5960000157356262},{"id":"https://openalex.org/keywords/perfect-powers","display_name":"perfect powers","score":0.6129999756813049},{"id":"https://openalex.org/keywords/diophantine-equations","display_name":"Diophantine equations","score":0.722000002861023},{"id":"https://openalex.org/keywords/generalized-riemann-hypothesis","display_name":"generalized Riemann hypothesis","score":0.8820000290870667},{"id":"https://openalex.org/keywords/dedekind-zeta-function","display_name":"Dedekind zeta function","score":0.9990000128746033},{"id":"https://openalex.org/keywords/cyclotomic-polynomials","display_name":"cyclotomic polynomials","score":0.7630000114440918},{"id":"https://openalex.org/keywords/explicit-formula","display_name":"explicit formula","score":0.7910000085830688},{"id":"https://openalex.org/keywords/valuation","display_name":"valuation","score":0.6510000228881836}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8978999853134155},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.7745000123977661},{"id":"https://openalex.org/C2780388253","wikidata":"https://www.wikidata.org/wiki/Q5421508","display_name":"Exponent","level":2,"score":0.701200008392334},{"id":"https://openalex.org/C183763347","wikidata":"https://www.wikidata.org/wiki/Q120976","display_name":"Factorial","level":2,"score":0.6427000164985657},{"id":"https://openalex.org/C203492994","wikidata":"https://www.wikidata.org/wiki/Q909669","display_name":"Divisor (algebraic geometry)","level":2,"score":0.5949000120162964},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5415999889373779},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4205000102519989},{"id":"https://openalex.org/C199479865","wikidata":"https://www.wikidata.org/wiki/Q205966","display_name":"Riemann hypothesis","level":2,"score":0.4074000120162964},{"id":"https://openalex.org/C30860621","wikidata":"https://www.wikidata.org/wiki/Q49008","display_name":"Prime factor","level":3,"score":0.3822000026702881},{"id":"https://openalex.org/C135692309","wikidata":"https://www.wikidata.org/wiki/Q111124","display_name":"Square (algebra)","level":2,"score":0.3528999984264374},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.3352000117301941},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.33000001311302185},{"id":"https://openalex.org/C143732363","wikidata":"https://www.wikidata.org/wiki/Q485520","display_name":"Goldbach's conjecture","level":3,"score":0.31540000438690186},{"id":"https://openalex.org/C42058472","wikidata":"https://www.wikidata.org/wiki/Q810214","display_name":"Base (topology)","level":2,"score":0.3050999939441681},{"id":"https://openalex.org/C76449584","wikidata":"https://www.wikidata.org/wiki/Q851333","display_name":"Dedekind cut","level":2,"score":0.3000999987125397},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.29350000619888306},{"id":"https://openalex.org/C2778067643","wikidata":"https://www.wikidata.org/wiki/Q166507","display_name":"Interval (graph theory)","level":2,"score":0.29159998893737793},{"id":"https://openalex.org/C125605598","wikidata":"https://www.wikidata.org/wiki/Q306393","display_name":"abc conjecture","level":4,"score":0.28870001435279846},{"id":"https://openalex.org/C169654258","wikidata":"https://www.wikidata.org/wiki/Q12479","display_name":"Number theory","level":2,"score":0.2874000072479248},{"id":"https://openalex.org/C174072685","wikidata":"https://www.wikidata.org/wiki/Q1667469","display_name":"Prime power","level":3,"score":0.27630001306533813}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.25661","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.25661","pdf_url":"https://arxiv.org/pdf/2610.25661","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.2610.25661","doi":"https://doi.org/10.48550/arxiv.2610.25661","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.25661","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"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.25661","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.25661","pdf_url":"https://arxiv.org/pdf/2610.25661","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/W7223012397.pdf","grobid_xml":"https://content.openalex.org/works/W7223012397.grobid-xml"},"referenced_works_count":4,"referenced_works":["https://openalex.org/W1522344093","https://openalex.org/W2296657975","https://openalex.org/W3043674320","https://openalex.org/W4233185505"],"related_works":["https://openalex.org/W78241493","https://openalex.org/W2155840112","https://openalex.org/W4299687838","https://openalex.org/W2803244953","https://openalex.org/W2086845540","https://openalex.org/W1968827684","https://openalex.org/W1037802398","https://openalex.org/W1787749081","https://openalex.org/W2041496871","https://openalex.org/W2074530356"],"abstract_inverted_index":{"We":[0,36],"propose":[1],"the":[2,33,38,60,65,78,108,151],"Factorial":[3],"Separation":[4],"Conjecture":[5],"(FSC),":[6],"a":[7,114],"strict":[8],"strengthening":[9],"of":[10,64,117],"earlier":[11],"factorial":[12,34],"gap":[13],"conjectures:":[14],"\\[":[15],"|n!-m^k|>n":[16],"\\]":[17],"for":[18,41,46,53,96,113],"all":[19,84],"integers":[20],"$n\\ge8$":[21,98],"and":[22,31,52,83,99,142],"$m,k\\ge2$.":[23],"The":[24,94,132,153],"inequality":[25],"excludes":[26],"every":[27,54,97,100],"displacement":[28],"up":[29],"to":[30],"including":[32],"index.":[35],"prove":[37,77],"conjecture":[39,155],"unconditionally":[40],"exponents":[42,48,72],"divisible":[43,73,102],"by":[44,74,103],"three,":[45],"even":[47],"at":[49,56,68],"least":[50,57],"four,":[51],"exponent":[55,101],"two":[58],"when":[59],"largest":[61],"prime":[62],"divisor":[63],"base":[66],"is":[67,105],"most":[69],"$n+2$.":[70],"For":[71],"five,":[75],"we":[76],"finite":[79,146],"interval":[80],"$8\\le":[81],"n\\le1100$":[82],"sufficiently":[85],"large":[86],"indices,":[87],"with":[88],"an":[89],"effective":[90],"but":[91],"uncomputed":[92],"threshold.":[93],"conclusion":[95],"five":[104],"proved":[106],"under":[107],"Generalized":[109],"Riemann":[110],"Hypothesis":[111],"(GRH)":[112],"specified":[115],"family":[116],"Dedekind":[118],"zeta":[119],"functions.":[120],"Here":[121],"GRH":[122],"asserts":[123],"that":[124],"their":[125],"nontrivial":[126],"zeros":[127],"have":[128],"real":[129],"part":[130],"$1/2$.":[131],"arguments":[133],"combine":[134],"valuations,":[135],"cyclotomic":[136],"factorizations,":[137],"cubic":[138],"reciprocity,":[139],"character":[140],"estimates":[141],"explicit":[143],"formulas.":[144],"All":[145],"comparisons":[147],"are":[148],"justified":[149],"in":[150],"text.":[152],"general":[154],"remains":[156],"open.":[157]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-21T00:00:00"}