History of Pisot numbers less than the golden ratio

 Display properties:   number-header: $\theta_r$+Numbers:+- params:+    r: '1'+  number: '1.324717957244746025960908854478097340734404056901733364534015050302827851245547594054699347981787280'+  comment: $P_{2}$ family; minimal polynomial $x^{3} - x - 1$; this is the plastic+    ratio.+- params:+    r: '2'+  number: '1.380277569097614115673301691822731877816626701558763025411771331211249574118641526187864568249035509'+  comment: $Q_{2}$ family; minimal polynomial $x^{4} - x^{3} - 1$.+- params:+    r: '3'+  number: '1.443268791270373107628127607386911604676011966654571598409233793623784837874189050037590075664743962'+  comment: $Q_{3}$ family; minimal polynomial $x^{5} - x^{4} - x^{3} + x^{2} - 1$.+- params:+    r: '4'+  number: '1.465571231876768026656731225219939108025577568472285701643183111249262996685017840478125801194909270'+  comment: $P_{3}$ family; minimal polynomial $x^{3} - x^{2} - 1$; this is the supergolden+    ratio.+- params:+    r: '5'+  number: '1.501594803539087366377783127371046108486398336253585342248394186063343612597128898134114246029200202'+  comment: $Q_{4}$ family; minimal polynomial $x^{6} - x^{5} - x^{4} + x^{2} - 1$.+- params:+    r: '6'+  number: '1.534157744914266915435970076109375701882545038516595135368531863008063023214082281436789664835483494'+  comment: $P_{4}$ family; minimal polynomial $x^{5} - x^{3} - x^{2} - x - 1$.+- params:+    r: '7'+  number: '1.545215649732755243252550624105116119691470055364233123560610725498211588166533120504318279380159156'+  comment: $Q_{5}$ family; minimal polynomial $x^{7} - x^{6} - x^{5} + x^{2} - 1$.+- params:+    r: '8'+  number: '1.561752067720297294702995364060723780790847286947276642846284783946252241043942944496244054750817375'+  comment: exceptional Dufresnoy-Pisot polynomial; minimal polynomial $x^{6} - 2 x^{5}+    + x^{4} - x^{2} + x - 1$.+- params:+    r: '9'+  number: '1.570147312196054362910665435137126553873131607424527436931654877897330661544162320222760040702620119'+  comment: $P_{5}$ family; minimal polynomial $x^{5} - x^{4} - x^{2} - 1$.+- params:+    r: '10'+  number: '1.573678968393516988774251418629321467812704061507913408937274370051211297448790471881548837806219190'+  comment: $Q_{6}$ family; minimal polynomial $x^{8} - x^{7} - x^{6} + x^{2} - 1$.+- params:+    r: '11'+  number: '1.590005373901363925162015541663808196898573270019121975693623393697501137164006258637372010170673034'+  comment: $P_{6}$ family; minimal polynomial $x^{7} - x^{5} - x^{4} - x^{3} - x^{2}+    - x - 1$.+- params:+    r: '12'+  number: '1.591184305667102506280378163106102721323455556070427118636105671632990046340380662184687387274732566'+  comment: $Q_{7}$ family; minimal polynomial $x^{9} - x^{8} - x^{7} + x^{2} - 1$.+- params:+    r: '13'+  number: '1.601347333787636724232618164313101117665668726676556628193381197227260290102478499805029985176068445'+  comment: $P_{7}$ family; minimal polynomial $x^{7} - x^{6} - x^{4} - x^{2} - 1$.+- params:+    r: '14'+  number: '1.601755861696983255736192654219002170232459847383645956556081860539331501794067175848485666834280654'+  comment: $Q_{8}$ family; minimal polynomial $x^{10} - x^{9} - x^{8} + x^{2} - 1$.+- params:+    r: '15'+  number: '1.607982727928201149922420457795797515362209291893843383821724387553882342235678008938055023076222152'+  comment: $P_{8}$ family; minimal polynomial $x^{9} - x^{7} - x^{6} - x^{5} - x^{4}+    - x^{3} - x^{2} - x - 1$.+- params:+    r: '16'+  number: '1.608128385187386959402200323538480539244506229939808475627654880454135115549049193244125291977586775'+  comment: $Q_{9}$ family; minimal polynomial $x^{11} - x^{10} - x^{9} + x^{2} - 1$.+- params:+    r: '17'+  number: '1.611930396564119819833599200722692620452821331844328319226299635293730595602174524805609485749220372'+  comment: $P_{9}$ family; minimal polynomial $x^{9} - x^{8} - x^{6} - x^{4} - x^{2}+    - 1$.+- params:+    r: '18'+  number: '1.611983421246492155859040049756323314852853437636639613720385059950750148002686762196644934656348672'+  comment: $Q_{10}$ family; minimal polynomial $x^{12} - x^{11} - x^{10} + x^{2} -+    1$.+- params:+    r: '19'+  number: '1.614306823257148514569632339911304106948171748039112819042361616990015585866967144509566552856428042'+  comment: $P_{10}$ family; minimal polynomial $x^{11} - x^{9} - x^{8} - x^{7} - x^{6}+    - x^{5} - x^{4} - x^{3} - x^{2} - x - 1$.+- params:+    r: '20'+  number: '1.614326414939127104073078356579636391949011342705773229044712283319374524946453994526083157310377742'+  comment: $Q_{11}$ family; minimal polynomial $x^{13} - x^{12} - x^{11} + x^{2} -+    1$.+- params:+    r: '21'+  number: '1.615749202755210610743641798789103362281243911263783155770251095174096083205827141199474172669686514'+  comment: $P_{11}$ family; minimal polynomial $x^{11} - x^{10} - x^{8} - x^{6} -+    x^{4} - x^{2} - 1$.+- params:+    r: '22'+  number: '1.615756517540843375529686781971131464046866598458441577197107007695607669818877612432159677719427740'+  comment: $Q_{12}$ family; minimal polynomial $x^{14} - x^{13} - x^{12} + x^{2} -+    1$.+- params:+    r: '23'+  number: '1.616629684394572703598804670579966020060105074535041910104107096102528067389296930735657559212555402'+  comment: $P_{12}$ family; minimal polynomial $x^{13} - x^{11} - x^{10} - x^{9} -+    x^{8} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x - 1$.+- params:+    r: '24'+  number: '1.616632435387905008141467676752220700831348754409358539038135095071601155042700459931124659244537941'+  comment: $Q_{13}$ family; minimal polynomial $x^{15} - x^{14} - x^{13} + x^{2} -+    1$.+- params:+    r: '25'+  number: '1.617169296355092563480671132600765362456967191135302102729539392961745062210998433921295220014506215'+  comment: $P_{13}$ family; minimal polynomial $x^{13} - x^{12} - x^{10} - x^{8} -+    x^{6} - x^{4} - x^{2} - 1$.+- params:+    r: '26'+  number: '1.617170336172016847605481712611059501452437635469882386057255508606235491186675769348455132093009414'+  comment: $Q_{14}$ family; minimal polynomial $x^{16} - x^{15} - x^{14} + x^{2} -+    1$.+- params:+    r: '27'+  number: '1.617500905431324014411963007650991447867821260857169838924218729322581246942327161872141255472644582'+  comment: $P_{14}$ family; minimal polynomial $x^{15} - x^{13} - x^{12} - x^{11}+    - x^{10} - x^{9} - x^{8} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x -+    1$.+- params:+    r: '28'+  number: '1.617501299812909557233137493529053056922071157913856951616026650849537504285172555316810694539178691'+  comment: $Q_{15}$ family; minimal polynomial $x^{17} - x^{16} - x^{15} + x^{2} -+    1$.+- params:+    r: '29'+  number: '1.617705069957556644470177554481987140361426228819198259427878209593425193281904169829510686690774651'+  comment: $P_{15}$ family; minimal polynomial $x^{15} - x^{14} - x^{12} - x^{10}+    - x^{8} - x^{6} - x^{4} - x^{2} - 1$.+- params:+    r: '30'+  number: '1.617705219888455097146323069204504687107702709103938695316495370965365042281749410718148250134450673'+  comment: $Q_{16}$ family; minimal polynomial $x^{18} - x^{17} - x^{16} + x^{2} -+    1$.+- params:+    r: '31'+  number: '1.617830928788973863699424746173140971640051423343242480743578861627018039776784999830916329455401512'+  comment: $P_{16}$ family; minimal polynomial $x^{17} - x^{15} - x^{14} - x^{13}+    - x^{12} - x^{11} - x^{10} - x^{9} - x^{8} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3}+    - x^{2} - x - 1$.+- params:+    r: '32'+  number: '1.617830985877812298846657162479186429296004329244380834320484689516026034285121093064090212646902933'+  comment: $Q_{17}$ family; minimal polynomial $x^{19} - x^{18} - x^{17} + x^{2} -+    1$.+- params:+    r: '33'+  number: '1.617908581767165011975654251943379527724351187695218752887992225503193582039747807291459082094368585'+  comment: $P_{17}$ family; minimal polynomial $x^{17} - x^{16} - x^{14} - x^{12}+    - x^{10} - x^{8} - x^{6} - x^{4} - x^{2} - 1$.+- params:+    r: '34'+  number: '1.617908603527805385791372655209449553767018177375841206557773359477283053340186177065247280643988320'+  comment: $Q_{18}$ family; minimal polynomial $x^{20} - x^{19} - x^{18} + x^{2} -+    1$.+- params:+    r: '35'+  number: '1.617956519953564239208622399882582744059606376335285602101209468687677811493689474942744891506041932'+  comment: $P_{18}$ family; minimal polynomial $x^{19} - x^{17} - x^{16} - x^{15}+    - x^{14} - x^{13} - x^{12} - x^{11} - x^{10} - x^{9} - x^{8} - x^{7} - x^{6} -+    x^{5} - x^{4} - x^{3} - x^{2} - x - 1$.+- params:+    r: '36'+  number: '1.617956528253976570213724233669342650311982524814136763277345754456220908746937624048581615827138001'+  comment: $Q_{19}$ family; minimal polynomial $x^{21} - x^{20} - x^{19} + x^{2} -+    1$.+- params:+    r: '37'+  number: '1.617986125385249151581179065219836276338293750679622676312918048191078288185448939191521385866321094'+  comment: $P_{19}$ family; minimal polynomial $x^{19} - x^{18} - x^{16} - x^{14}+    - x^{12} - x^{10} - x^{8} - x^{6} - x^{4} - x^{2} - 1$.+- params:+    r: '38'+  number: '1.617986128552861828746775573560971560626559554179903857177524556107337154595403717855874656309277088'+  comment: $Q_{20}$ family; minimal polynomial $x^{22} - x^{21} - x^{20} + x^{2} -+    1$.+- params:+    r: '39'+  number: '1.618004413617124503253339884205705740949886990727485864190374404055016480918880509307231034414017485'+  comment: $P_{20}$ family; minimal polynomial $x^{21} - x^{19} - x^{18} - x^{17}+    - x^{16} - x^{15} - x^{14} - x^{13} - x^{12} - x^{11} - x^{10} - x^{9} - x^{8}+    - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x - 1$.+- params:+    r: '40'+  number: '1.618004414826328965224816813048886874741852358452825536420120516315769101498118708366546047544102188'+  comment: $Q_{21}$ family; minimal polynomial $x^{23} - x^{22} - x^{21} + x^{2} -+    1$.+- params:+    r: '41'+  number: '1.618015712751258280635900473733667949090056927598243987323992158078160732484712147251547557149689392'+  comment: $P_{21}$ family; minimal polynomial $x^{21} - x^{20} - x^{18} - x^{16}+    - x^{14} - x^{12} - x^{10} - x^{8} - x^{6} - x^{4} - x^{2} - 1$.+- params:+    r: '42'+  number: '1.618015713212954752257255964375369229374123216484787415640736393798376918798139605181967819732712222'+  comment: $Q_{22}$ family; minimal polynomial $x^{24} - x^{23} - x^{22} + x^{2} -+    1$.+- params:+    r: '43'+  number: '1.618022694541460339603743229827892343352486377764735927967607280632891128415448447893953664301512573'+  comment: $P_{22}$ family; minimal polynomial $x^{23} - x^{21} - x^{20} - x^{19}+    - x^{18} - x^{17} - x^{16} - x^{15} - x^{14} - x^{13} - x^{12} - x^{11} - x^{10}+    - x^{9} - x^{8} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x - 1$.+- params:+    r: '44'+  number: '1.618022694717768294416502444185158998331096020868084111924521311236631268730313972376491207972498484'+  comment: $Q_{23}$ family; minimal polynomial $x^{25} - x^{24} - x^{23} + x^{2} -+    1$.+- params:+    r: '45'+  number: '1.618027008938025718553681758942751295768660974229159521330823899915592183362260151838715294021339562'+  comment: $P_{23}$ family; minimal polynomial $x^{23} - x^{22} - x^{20} - x^{18}+    - x^{16} - x^{14} - x^{12} - x^{10} - x^{8} - x^{6} - x^{4} - x^{2} - 1$.+- params:+    r: '46'+  number: '1.618027009005358347427925536092720504146670868386797655664278670383537383085953832521182612767970098'+  comment: $Q_{24}$ family; minimal polynomial $x^{26} - x^{25} - x^{24} + x^{2} -+    1$.+- params:+    r: '47'+  number: '1.618029675146055305698946786118197148936629694393732406286549986112999293588968023123008043461804993'+  comment: $P_{24}$ family; minimal polynomial $x^{25} - x^{23} - x^{22} - x^{21}+    - x^{20} - x^{19} - x^{18} - x^{17} - x^{16} - x^{15} - x^{14} - x^{13} - x^{12}+    - x^{11} - x^{10} - x^{9} - x^{8} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2}+    - x - 1$.+- params:+    r: '48'+  number: '1.618029675171771354015094432833994648403067615748015040807374613270327489817157619418160494014043687'+  comment: $Q_{25}$ family; minimal polynomial $x^{27} - x^{26} - x^{25} + x^{2} -+    1$.+- params:+    r: '49'+  number: '1.618031322858835939379993750912293610423937558487693101083392560992438305802819245773759782326311482'+  comment: $P_{25}$ family; minimal polynomial $x^{25} - x^{24} - x^{22} - x^{20}+    - x^{18} - x^{16} - x^{14} - x^{12} - x^{10} - x^{8} - x^{6} - x^{4} - x^{2} -+    1$.+- params:+    r: '50'+  number: '1.618031322868657922029597723248317005726758817620183761258070106896742787251271592809502788045381653'+  comment: $Q_{26}$ family; minimal polynomial $x^{28} - x^{27} - x^{26} + x^{2} -+    1$.