{"id":"https://openalex.org/W7204561695","doi":"https://doi.org/10.48550/arxiv.2608.26384","title":"Discrete dynamics of Eratosthenes sieve","display_name":"Discrete dynamics of Eratosthenes sieve","publication_year":2026,"publication_date":"2026-08-26","ids":{"openalex":"https://openalex.org/W7204561695","doi":"https://doi.org/10.48550/arxiv.2608.26384"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2608.26384","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.26384","pdf_url":"https://arxiv.org/pdf/2608.26384","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/2608.26384","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5113014566","display_name":"Fred B. Holt","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Holt, Fred B.","raw_affiliation_strings":["Seattle , WA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Seattle , WA","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5113014566"],"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":0.9990000128746033,"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":0.9990000128746033,"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/T12516","display_name":"Advanced Mathematical Theories and Applications","score":9.999999747378752e-05,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"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":9.999999747378752e-05,"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/sieve-of-eratosthenes","display_name":"sieve of Eratosthenes","score":0.972000002861023},{"id":"https://openalex.org/keywords/prime-gaps","display_name":"prime gaps","score":0.9470000267028809},{"id":"https://openalex.org/keywords/prime-constellations","display_name":"prime constellations","score":0.7739999890327454},{"id":"https://openalex.org/keywords/markov-chains","display_name":"Markov chains","score":0.8119999766349792},{"id":"https://openalex.org/keywords/prime-k-tuples","display_name":"prime k-tuples","score":0.597000002861023}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7732999920845032},{"id":"https://openalex.org/C2908647359","wikidata":"https://www.wikidata.org/wiki/Q2625603","display_name":"Population","level":2,"score":0.5792999863624573},{"id":"https://openalex.org/C2778067643","wikidata":"https://www.wikidata.org/wiki/Q166507","display_name":"Interval (graph theory)","level":2,"score":0.5421000123023987},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.44020000100135803},{"id":"https://openalex.org/C168773036","wikidata":"https://www.wikidata.org/wiki/Q264164","display_name":"Recursion (computer science)","level":2,"score":0.4343000054359436},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.4075999855995178},{"id":"https://openalex.org/C98763669","wikidata":"https://www.wikidata.org/wiki/Q176645","display_name":"Markov chain","level":2,"score":0.4004000127315521},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.390500009059906},{"id":"https://openalex.org/C55689738","wikidata":"https://www.wikidata.org/wiki/Q15963867","display_name":"Discrete time and continuous time","level":2,"score":0.37119999527931213},{"id":"https://openalex.org/C2778753569","wikidata":"https://www.wikidata.org/wiki/Q1960395","display_name":"Span (engineering)","level":2,"score":0.36910000443458557},{"id":"https://openalex.org/C159886148","wikidata":"https://www.wikidata.org/wiki/Q176645","display_name":"Markov process","level":2,"score":0.36739999055862427},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.3668000102043152},{"id":"https://openalex.org/C17732976","wikidata":"https://www.wikidata.org/wiki/Q7511782","display_name":"Sieve (category theory)","level":2,"score":0.3407000005245209},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.3287999927997589},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.3264000117778778},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.3156000077724457},{"id":"https://openalex.org/C155512373","wikidata":"https://www.wikidata.org/wiki/Q287450","display_name":"Residual","level":2,"score":0.2870999872684479},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.2856999933719635},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.28459998965263367},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.28200000524520874},{"id":"https://openalex.org/C75438885","wikidata":"https://www.wikidata.org/wiki/Q3403615","display_name":"Large deviations theory","level":2,"score":0.2757999897003174},{"id":"https://openalex.org/C57493831","wikidata":"https://www.wikidata.org/wiki/Q3134666","display_name":"Projection (relational algebra)","level":2,"score":0.27390000224113464},{"id":"https://openalex.org/C89128539","wikidata":"https://www.wikidata.org/wiki/Q1949963","display_name":"Statistic","level":2,"score":0.2694000005722046},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.2671999931335449},{"id":"https://openalex.org/C52079815","wikidata":"https://www.wikidata.org/wiki/Q7229808","display_name":"Population model","level":3,"score":0.26350000500679016},{"id":"https://openalex.org/C49937458","wikidata":"https://www.wikidata.org/wiki/Q2599292","display_name":"Probabilistic logic","level":2,"score":0.26159998774528503},{"id":"https://openalex.org/C70352696","wikidata":"https://www.wikidata.org/wiki/Q8928","display_name":"Constellation","level":2,"score":0.2603999972343445},{"id":"https://openalex.org/C113429393","wikidata":"https://www.wikidata.org/wiki/Q49008","display_name":"Prime number","level":2,"score":0.2563999891281128}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2608.26384","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.26384","pdf_url":"https://arxiv.org/pdf/2608.26384","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.2608.26384","doi":"https://doi.org/10.48550/arxiv.2608.26384","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2608.26384","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:2608.26384","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.26384","pdf_url":"https://arxiv.org/pdf/2608.26384","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/W7204561695.pdf","grobid_xml":"https://content.openalex.org/works/W7204561695.grobid-xml"},"referenced_works_count":4,"referenced_works":["https://openalex.org/W2196468737","https://openalex.org/W4387293314","https://openalex.org/W4396788210","https://openalex.org/W6941510939"],"related_works":["https://openalex.org/W7219144568","https://openalex.org/W7143345831","https://openalex.org/W1837106074","https://openalex.org/W4387293314","https://openalex.org/W2196468737","https://openalex.org/W134211640","https://openalex.org/W6979261245","https://openalex.org/W1702240654","https://openalex.org/W2159960217","https://openalex.org/W2336977718"],"abstract_inverted_index":{"We":[0,221,237],"study":[1],"Eratosthenes":[2],"sieve":[3,14],"as":[4,109],"a":[5,17,31,165,180,189,223],"discrete":[6],"dynamic":[7],"system.":[8],"At":[9],"each":[10],"stage":[11],"of":[12,19,23,60,73,92,101,130,138,160,188,203,206,215,229],"the":[13,39,43,52,70,85,93,117,122,126,131,139,161,174,186,204,213,226,230,234,240],"there":[15],"is":[16,30,97,164],"cycle":[18,41,53],"gaps":[20,66,246],"${\\mathcal":[21,33,54,197],"G}(p^\\#)$":[22],"length":[24,102],"$\u03d5(p^\\#)$":[25],"and":[26],"span":[27,61,181],"$p^\\#$.":[28],"There":[29],"recursion":[32],"G}(p_k^\\#)\\longrightarrow":[34],"{\\mathcal":[35],"G}(p_{k+1}^\\#)$":[36],"that":[37,170,178,185,209,239],"creates":[38],"next":[40],"from":[42,51],"current":[44],"one.":[45],"If":[46,95],"we":[47,114,199],"take":[48],"initial":[49,201,241],"conditions":[50],"G}(p_0^\\#)$,":[55],"then":[56,104],"for":[57,84,88,125,244],"all":[58,89,135],"constellations":[59],"$|s|":[62],"<":[63,68,151],"2p_1$,":[64,69],"including":[65],"$g":[67],"driving":[71],"terms":[72],"various":[74],"lengths":[75],"form":[76],"Markov":[77],"chains.":[78],"These":[79],"yield":[80],"{\\em":[81],"exact}":[82],"models":[83],"populations":[86],"$n_s(p_k^\\#)$":[87],"further":[90,136],"stages":[91,137],"sieve.":[94,140],"$s$":[96,133,191,208,232],"an":[98],"admissible":[99],"constellation":[100,132,190,231],"$J$,":[103],"its":[105],"population":[106,128,163],"$n_{s,J}(p^\\#)$":[107],"grows":[108],"$\u0398\\left(":[110],"\\prod":[111],"(q-J-1)\\right)$.":[112],"So":[113],"factor":[115],"out":[116],"superexponential":[118],"growth":[119],"to":[120,251],"obtain":[121],"exact":[123],"model":[124],"relative":[127,162],"$w_s(p_k^\\#)$":[129],"across":[134],"$$":[141,156],"w_{s,J}(p_k^\\#)":[142],"\\;":[143,145],"=":[144,218],"n_{s,J}(p_k^\\#)":[146],"\\,":[147,149],"/":[148],"\\prod_{J+1":[150],"p":[152],"\\le":[153],"p_k}":[154],"(p-J-1)":[155],"The":[157],"asymptotic":[158],"value":[159],"constant":[166],"${w_{s,J}(\\infty)":[167],"\\ge":[168],"1}$":[169],"depends":[171],"only":[172],"on":[173],"odd":[175],"prime":[176,245],"factors":[177],"divide":[179],"in":[182,196,212],"$s$.":[183],"Assuming":[184],"instances":[187,207],"are":[192],"approximately":[193],"uniformly":[194],"distributed":[195],"G}(p_k^\\#)$,":[198],"develop":[200],"estimates":[202,242],"number":[205],"would":[210],"occur":[211],"interval":[214,235],"survival":[216],"$\u0394H(p_k)":[217],"(p_k^2,":[219],"p_{k+1}^2)$.":[220],"define":[222],"statistic":[224],"$\u03b7_s(p_k)$,":[225],"quadratic":[227],"density":[228],"over":[233],"$\u0394H(p_k)$.":[236],"show":[238],"$\\widehat{\u03b7_g}(p)$":[243],"agree":[247],"with":[248],"samples":[249],"up":[250],"$5.677\\,E14$.":[252]},"counts_by_year":[],"updated_date":"2026-11-08T12:12:28.789561","created_date":"2026-08-29T00:00:00"}