{"id":"https://openalex.org/W2145880011","doi":"https://doi.org/10.1080/10586458.2002.10504704","title":"Random Generators and Normal Numbers","display_name":"Random Generators and Normal Numbers","publication_year":2002,"publication_date":"2002-01-01","ids":{"openalex":"https://openalex.org/W2145880011","doi":"https://doi.org/10.1080/10586458.2002.10504704","mag":"2145880011"},"language":"en","primary_location":{"id":"doi:10.1080/10586458.2002.10504704","is_oa":true,"landing_page_url":"https://doi.org/10.1080/10586458.2002.10504704","pdf_url":"https://www.tandfonline.com/doi/abs/10.1080/10586458.2002.10504704","source":{"id":"https://openalex.org/S119636414","display_name":"Experimental Mathematics","issn_l":"1058-6458","issn":["1058-6458","1944-950X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547","https://openalex.org/P4310320449"],"host_organization_lineage_names":["Taylor & Francis","Informa"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Experimental Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://www.tandfonline.com/doi/abs/10.1080/10586458.2002.10504704","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5081580336","display_name":"David H. Bailey","orcid":"https://orcid.org/0000-0002-7574-8342"},"institutions":[{"id":"https://openalex.org/I148283060","display_name":"Lawrence Berkeley National Laboratory","ror":"https://ror.org/02jbv0t02","country_code":"US","type":"facility","lineage":["https://openalex.org/I1330989302","https://openalex.org/I148283060","https://openalex.org/I39565521"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"David H. Bailey","raw_affiliation_strings":["Lawrence Berkeley National Laboratory, Berkeley, CA, 94720 E-mail: dhbailey@lbl.gov","Lawrence Berkeley National Laboratory , Berkeley , CA , 94720 E-mail: dhbailey@lbl.gov"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Lawrence Berkeley National Laboratory, Berkeley, CA, 94720 E-mail: dhbailey@lbl.gov","institution_ids":["https://openalex.org/I148283060"]},{"raw_affiliation_string":"Lawrence Berkeley National Laboratory , Berkeley , CA , 94720 E-mail: dhbailey@lbl.gov","institution_ids":["https://openalex.org/I148283060"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5111508743","display_name":"Richard E. Crandall","orcid":null},"institutions":[{"id":"https://openalex.org/I55486353","display_name":"Reed College","ror":"https://ror.org/00a6ram87","country_code":"US","type":"education","lineage":["https://openalex.org/I55486353"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Richard E. Crandall","raw_affiliation_strings":["Center for Advanced Computation, Reed College","Center for Advanced Computation, Reed College , Portland , OR , 97202 E-mail: crandall@reed.edu"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Center for Advanced Computation, Reed College","institution_ids":["https://openalex.org/I55486353"]},{"raw_affiliation_string":"Center for Advanced Computation, Reed College , Portland , OR , 97202 E-mail: crandall@reed.edu","institution_ids":["https://openalex.org/I55486353"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":4.5011,"has_fulltext":true,"cited_by_count":90,"citation_normalized_percentile":{"value":0.95126156,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"11","issue":"4","first_page":"527","last_page":"546"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11017","display_name":"Chaos-based Image/Signal Encryption","score":0.9990000128746033,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11017","display_name":"Chaos-based Image/Signal Encryption","score":0.9990000128746033,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.9975000023841858,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.9972000122070312,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.852726936340332},{"id":"https://openalex.org/keywords/pseudorandom-number-generator","display_name":"Pseudorandom number generator","score":0.7800878286361694},{"id":"https://openalex.org/keywords/coprime-integers","display_name":"Coprime integers","score":0.605201244354248},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.5381219387054443},{"id":"https://openalex.org/keywords/rational-number","display_name":"Rational number","score":0.4809812307357788},{"id":"https://openalex.org/keywords/constant","display_name":"Constant (computer programming)","score":0.4652479588985443},{"id":"https://openalex.org/keywords/uncountable-set","display_name":"Uncountable set","score":0.44775670766830444},{"id":"https://openalex.org/keywords/class","display_name":"Class (philosophy)","score":0.4438363015651703},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.41369181871414185},{"id":"https://openalex.org/keywords/countable-set","display_name":"Countable set","score":0.3467627167701721},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.32513123750686646},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.11868688464164734}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.852726936340332},{"id":"https://openalex.org/C140642157","wikidata":"https://www.wikidata.org/wiki/Q1623338","display_name":"Pseudorandom number generator","level":2,"score":0.7800878286361694},{"id":"https://openalex.org/C23230895","wikidata":"https://www.wikidata.org/wiki/Q104752","display_name":"Coprime integers","level":2,"score":0.605201244354248},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5381219387054443},{"id":"https://openalex.org/C46875033","wikidata":"https://www.wikidata.org/wiki/Q1244890","display_name":"Rational number","level":2,"score":0.4809812307357788},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.4652479588985443},{"id":"https://openalex.org/C142399903","wikidata":"https://www.wikidata.org/wiki/Q1128796","display_name":"Uncountable set","level":3,"score":0.44775670766830444},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.4438363015651703},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.41369181871414185},{"id":"https://openalex.org/C110729354","wikidata":"https://www.wikidata.org/wiki/Q185478","display_name":"Countable set","level":2,"score":0.3467627167701721},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.32513123750686646},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.11868688464164734},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.1080/10586458.2002.10504704","is_oa":true,"landing_page_url":"https://doi.org/10.1080/10586458.2002.10504704","pdf_url":"https://www.tandfonline.com/doi/abs/10.1080/10586458.2002.10504704","source":{"id":"https://openalex.org/S119636414","display_name":"Experimental Mathematics","issn_l":"1058-6458","issn":["1058-6458","1944-950X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547","https://openalex.org/P4310320449"],"host_organization_lineage_names":["Taylor & Francis","Informa"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Experimental Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.107.3936","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.107.3936","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://crd.lbl.gov/~dhbailey/dhbpapers/bcnormal-em.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.23.9557","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.23.9557","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.nersc.gov/~dhb/dhbpapers/bcnormal.ps","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.9.6808","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.9.6808","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.nersc.gov/~dhbailey/dhbpapers/bcnormal.pdf","raw_type":"text"},{"id":"pmh:oai:CULeuclid:euclid.em/1057864662","is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.em/1057864662","pdf_url":null,"source":{"id":"https://openalex.org/S4306400787","display_name":"Project Euclid (Cornell University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"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":"","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.1080/10586458.2002.10504704","is_oa":true,"landing_page_url":"https://doi.org/10.1080/10586458.2002.10504704","pdf_url":"https://www.tandfonline.com/doi/abs/10.1080/10586458.2002.10504704","source":{"id":"https://openalex.org/S119636414","display_name":"Experimental Mathematics","issn_l":"1058-6458","issn":["1058-6458","1944-950X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547","https://openalex.org/P4310320449"],"host_organization_lineage_names":["Taylor & Francis","Informa"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Experimental Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1398330795","display_name":null,"funder_award_id":"DE-AC03-?","funder_id":"https://openalex.org/F4320306084","funder_display_name":"U.S. Department of Energy"},{"id":"https://openalex.org/G1643223109","display_name":null,"funder_award_id":"AC03-76SF00098","funder_id":"https://openalex.org/F4320306084","funder_display_name":"U.S. Department of Energy"},{"id":"https://openalex.org/G193590374","display_name":null,"funder_award_id":"DEAC03-76SF00098","funder_id":"https://openalex.org/F4320306084","funder_display_name":"U.S. Department of Energy"}],"funders":[{"id":"https://openalex.org/F4320306084","display_name":"U.S. Department of Energy","ror":"https://ror.org/01bj3aw27"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2145880011.pdf","grobid_xml":"https://content.openalex.org/works/W2145880011.grobid-xml"},"referenced_works_count":55,"referenced_works":["https://openalex.org/W870636097","https://openalex.org/W981720763","https://openalex.org/W1034323107","https://openalex.org/W1040474638","https://openalex.org/W1229049121","https://openalex.org/W1489341787","https://openalex.org/W1504894794","https://openalex.org/W1522031204","https://openalex.org/W1531131534","https://openalex.org/W1537715246","https://openalex.org/W1546385162","https://openalex.org/W1563654733","https://openalex.org/W1607458528","https://openalex.org/W1976927746","https://openalex.org/W1978915576","https://openalex.org/W1984582452","https://openalex.org/W2007343403","https://openalex.org/W2011080089","https://openalex.org/W2014937115","https://openalex.org/W2018462124","https://openalex.org/W2019338982","https://openalex.org/W2040087553","https://openalex.org/W2043419861","https://openalex.org/W2046623466","https://openalex.org/W2046937767","https://openalex.org/W2051221766","https://openalex.org/W2065534115","https://openalex.org/W2069328821","https://openalex.org/W2073085121","https://openalex.org/W2073394774","https://openalex.org/W2075049308","https://openalex.org/W2076597612","https://openalex.org/W2082345354","https://openalex.org/W2092337953","https://openalex.org/W2095332217","https://openalex.org/W2095709299","https://openalex.org/W2103236565","https://openalex.org/W2109304932","https://openalex.org/W2117495412","https://openalex.org/W2120579522","https://openalex.org/W2138039317","https://openalex.org/W2140262981","https://openalex.org/W2165306900","https://openalex.org/W2330231397","https://openalex.org/W2479395228","https://openalex.org/W2752061190","https://openalex.org/W2752853835","https://openalex.org/W3038928148","https://openalex.org/W3175367423","https://openalex.org/W4233220581","https://openalex.org/W4239269155","https://openalex.org/W4239648988","https://openalex.org/W4248483575","https://openalex.org/W4300914388","https://openalex.org/W4402905022"],"related_works":["https://openalex.org/W3019662020","https://openalex.org/W4297571715","https://openalex.org/W2100222222","https://openalex.org/W2512721618","https://openalex.org/W4242301765","https://openalex.org/W1519257018","https://openalex.org/W3139954723","https://openalex.org/W3097468110","https://openalex.org/W2328843190","https://openalex.org/W2480446702"],"abstract_inverted_index":{"Pursuant":[0],"to":[1,150,185],"the":[2,42,76,94,107,119,151,156,168,194],"authors'":[3],"previous":[4],"chaotic-dynamical":[5],"model":[6],"for":[7,39,49,70,106,117,178],"random":[8],"digits":[9,172],"of":[10,41,54,64,97,145,155,201,210],"fundamental":[11,161],"constants":[12,40,162],"[Bailey":[13],"and":[14,59,216],"Crandall":[15],"01],":[16],"we":[17,140,191],"investigate":[18],"a":[19,208],"complementary,":[20],"statistical":[21],"picture":[22],"in":[23],"which":[24,118],"pseudorandom":[25],"number":[26],"generators":[27],"(PRNGs)":[28],"are":[29,34],"central.":[30],"Some":[31],"rigorous":[32],"results":[33,215],"achieved:":[35],"We":[36,67,205],"establish":[37],"b-normality":[38,105],"form":[43],"\u03a3":[44,81],"1/(b":[45],"mi":[46],"c":[47,73,114],"ni)":[48],"certain":[50],"sequences":[51],"(mi),":[52],"(ni)":[53],"integers.":[55],"This":[56],"work":[57],"unifies":[58],"extends":[60],"previously":[61],"known":[62],"classes":[63,158,181],"explicit":[65,146],"normals.":[66],"prove":[68],"that":[69,148,170,193],"coprime":[71],"b,":[72,113],">":[74,137],"1":[75],"constant":[77],"\u03b1":[78],"b,c":[79],"=":[80],"n=c,c":[82],"2,c":[83],"3":[84],",":[85,116,129,131,133],"\u2026":[86,134],"1/(nb":[87],"n":[88,122],")":[89],"is":[90,203],"b-normal,":[91],"thus":[92],"generalizing":[93],"Stoneham":[95],"class":[96,109,144],"normals":[98,147],"[Stoneham":[99],"73a].":[100],"Our":[101],"approach":[102],"also":[103,206],"reproves":[104],"Korobov":[108],"[Korobov":[110],"90]":[111],"\u03b2":[112],"d":[115,136],"summation":[120],"index":[121],"above":[123],"runs":[124],"instead":[125],"over":[126],"powers":[127],"cd":[128],"cd2":[130],"cd3":[132],"with":[135,160],"1.":[138],"Eventually":[139],"describe":[141],"an":[142],"uncountable":[143],"succumb":[149],"PRNG":[152,221],"approach.":[153],"Numbers":[154],"\u03b1,\u03b2":[157],"share":[159],"such":[163,182],"as":[164,213],"\u03c0,":[165],"log":[166],"2":[167],"property":[169],"isolated":[171],"can":[173],"be":[174,186],"directly":[175],"calculated,":[176],"but":[177],"these":[179],"new":[180],"computation":[183],"tends":[184],"surprisingly":[187],"rapid.":[188],"For":[189],"example,":[190],"find":[192],"googol-th":[195],"(i.e.,":[196],"10100":[197],"-th)":[198],"binary":[199],"bit":[200],"\u03b12,3":[202],"0.":[204],"present":[207],"collection":[209],"other":[211],"results\u2014such":[212],"digit-density":[214],"irrationality":[217],"proofs":[218],"based":[219],"on":[220],"ideas\u2014for":[222],"various":[223],"special":[224],"numbers.":[225]},"counts_by_year":[{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":4},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":3},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":3},{"year":2016,"cited_by_count":4},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":5},{"year":2013,"cited_by_count":8},{"year":2012,"cited_by_count":8}],"updated_date":"2026-08-26T07:47:46.906454","created_date":"2025-10-10T00:00:00"}