Pisot numbers less than the golden ratio

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Numbers

$r$

$\theta_r$

1.324717957244746025960908854478097340734404056901733364534015050302827851245547594054699347981787280

comment: A root of $P_{2}$, with minimal polynomial $x^{3} - x - 1$. This is the plastic ratio, the smallest Pisot number. It is the growth rate of $\Delta(2,3,\infty)$.

1.380277569097614115673301691822731877816626701558763025411771331211249574118641526187864568249035509

comment: A root of $Q_{2}$, with minimal polynomial $x^{4} - x^{3} - 1$.

1.443268791270373107628127607386911604676011966654571598409233793623784837874189050037590075664743962

comment: A root of $Q_{3}$, with minimal polynomial $x^{5} - x^{4} - x^{3} + x^{2} - 1$.

1.465571231876768026656731225219939108025577568472285701643183111249262996685017840478125801194909270

comment: A root of $P_{3}$, with minimal polynomial $x^{3} - x^{2} - 1$. This is the supergolden ratio. It is the growth rate of $\Delta(2,4,\infty)$.

1.501594803539087366377783127371046108486398336253585342248394186063343612597128898134114246029200202

comment: A root of $Q_{4}$, with minimal polynomial $x^{6} - x^{5} - x^{4} + x^{2} - 1$.

1.534157744914266915435970076109375701882545038516595135368531863008063023214082281436789664835483494

comment: A root of $P_{4}$, with minimal polynomial $x^{5} - x^{3} - x^{2} - x - 1$. It is the growth rate of $\Delta(2,5,\infty)$.

1.545215649732755243252550624105116119691470055364233123560610725498211588166533120504318279380159156

comment: A root of $Q_{5}$, with minimal polynomial $x^{7} - x^{6} - x^{5} + x^{2} - 1$.

1.561752067720297294702995364060723780790847286947276642846284783946252241043942944496244054750817375

comment: The root in $(1,\varphi)$ of $E$, with minimal polynomial $x^{6} - 2x^{5} + x^{4} - x^{2} + x - 1$.

1.570147312196054362910665435137126553873131607424527436931654877897330661544162320222760040702620119

comment: A root of $P_{5}$, with minimal polynomial $x^{5} - x^{4} - x^{2} - 1$. It is the growth rate of $\Delta(2,6,\infty)$.

10:

1.573678968393516988774251418629321467812704061507913408937274370051211297448790471881548837806219190

comment: A root of $Q_{6}$, with minimal polynomial $x^{8} - x^{7} - x^{6} + x^{2} - 1$.

11:

1.590005373901363925162015541663808196898573270019121975693623393697501137164006258637372010170673034

comment: A root of $P_{6}$, with minimal polynomial $x^{7} - x^{5} - x^{4} - x^{3} - x^{2} - x - 1$. It is the growth rate of $\Delta(2,7,\infty)$.

12:

1.591184305667102506280378163106102721323455556070427118636105671632990046340380662184687387274732566

comment: A root of $Q_{7}$, with minimal polynomial $x^{9} - x^{8} - x^{7} + x^{2} - 1$.

13:

1.601347333787636724232618164313101117665668726676556628193381197227260290102478499805029985176068445

comment: A root of $P_{7}$, with minimal polynomial $x^{7} - x^{6} - x^{4} - x^{2} - 1$. It is the growth rate of $\Delta(2,8,\infty)$.

14:

1.601755861696983255736192654219002170232459847383645956556081860539331501794067175848485666834280654

comment: A root of $Q_{8}$, with minimal polynomial $x^{10} - x^{9} - x^{8} + x^{2} - 1$.

15:

1.607982727928201149922420457795797515362209291893843383821724387553882342235678008938055023076222152

comment: A root of $P_{8}$, with minimal polynomial $x^{9} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x - 1$. It is the growth rate of $\Delta(2,9,\infty)$.

16:

1.608128385187386959402200323538480539244506229939808475627654880454135115549049193244125291977586775

comment: A root of $Q_{9}$, with minimal polynomial $x^{11} - x^{10} - x^{9} + x^{2} - 1$.

17:

1.611930396564119819833599200722692620452821331844328319226299635293730595602174524805609485749220372

comment: A root of $P_{9}$, with minimal polynomial $x^{9} - x^{8} - x^{6} - x^{4} - x^{2} - 1$. It is the growth rate of $\Delta(2,10,\infty)$.

18:

1.611983421246492155859040049756323314852853437636639613720385059950750148002686762196644934656348672

comment: A root of $Q_{10}$, with minimal polynomial $x^{12} - x^{11} - x^{10} + x^{2} - 1$.

19:

1.614306823257148514569632339911304106948171748039112819042361616990015585866967144509566552856428042

comment: A root of $P_{10}$, with minimal polynomial $x^{11} - x^{9} - x^{8} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x - 1$. It is the growth rate of $\Delta(2,11,\infty)$.

20:

1.614326414939127104073078356579636391949011342705773229044712283319374524946453994526083157310377742

comment: A root of $Q_{11}$, with minimal polynomial $x^{13} - x^{12} - x^{11} + x^{2} - 1$.

21:

1.615749202755210610743641798789103362281243911263783155770251095174096083205827141199474172669686514

comment: A root of $P_{11}$, with minimal polynomial $x^{11} - x^{10} - x^{8} - x^{6} - x^{4} - x^{2} - 1$. It is the growth rate of $\Delta(2,12,\infty)$.

22:

1.615756517540843375529686781971131464046866598458441577197107007695607669818877612432159677719427740

comment: A root of $Q_{12}$, with minimal polynomial $x^{14} - x^{13} - x^{12} + x^{2} - 1$.

23:

1.616629684394572703598804670579966020060105074535041910104107096102528067389296930735657559212555402

comment: A root of $P_{12}$, with minimal polynomial $x^{13} - x^{11} - x^{10} - x^{9} - x^{8} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x - 1$.

24:

1.616632435387905008141467676752220700831348754409358539038135095071601155042700459931124659244537941

comment: A root of $Q_{13}$, with minimal polynomial $x^{15} - x^{14} - x^{13} + x^{2} - 1$.

25:

1.617169296355092563480671132600765362456967191135302102729539392961745062210998433921295220014506215

comment: A root of $P_{13}$, with minimal polynomial $x^{13} - x^{12} - x^{10} - x^{8} - x^{6} - x^{4} - x^{2} - 1$.

26:

1.617170336172016847605481712611059501452437635469882386057255508606235491186675769348455132093009414

comment: A root of $Q_{14}$, with minimal polynomial $x^{16} - x^{15} - x^{14} + x^{2} - 1$.

27:

1.617500905431324014411963007650991447867821260857169838924218729322581246942327161872141255472644582

comment: A root of $P_{14}$, with minimal polynomial $x^{15} - x^{13} - x^{12} - x^{11} - x^{10} - x^{9} - x^{8} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x - 1$.

28:

1.617501299812909557233137493529053056922071157913856951616026650849537504285172555316810694539178691

comment: A root of $Q_{15}$, with minimal polynomial $x^{17} - x^{16} - x^{15} + x^{2} - 1$.

29:

1.617705069957556644470177554481987140361426228819198259427878209593425193281904169829510686690774651

comment: A root of $P_{15}$, with minimal polynomial $x^{15} - x^{14} - x^{12} - x^{10} - x^{8} - x^{6} - x^{4} - x^{2} - 1$.

30:

1.617705219888455097146323069204504687107702709103938695316495370965365042281749410718148250134450673

comment: A root of $Q_{16}$, with minimal polynomial $x^{18} - x^{17} - x^{16} + x^{2} - 1$.

31:

1.617830928788973863699424746173140971640051423343242480743578861627018039776784999830916329455401512

comment: A root of $P_{16}$, with minimal polynomial $x^{17} - x^{15} - x^{14} - x^{13} - x^{12} - x^{11} - x^{10} - x^{9} - x^{8} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x - 1$.

32:

1.617830985877812298846657162479186429296004329244380834320484689516026034285121093064090212646902933

comment: A root of $Q_{17}$, with minimal polynomial $x^{19} - x^{18} - x^{17} + x^{2} - 1$.

33:

1.617908581767165011975654251943379527724351187695218752887992225503193582039747807291459082094368585

comment: A root of $P_{17}$, with minimal polynomial $x^{17} - x^{16} - x^{14} - x^{12} - x^{10} - x^{8} - x^{6} - x^{4} - x^{2} - 1$.

34:

1.617908603527805385791372655209449553767018177375841206557773359477283053340186177065247280643988320

comment: A root of $Q_{18}$, with minimal polynomial $x^{20} - x^{19} - x^{18} + x^{2} - 1$.

35:

1.617956519953564239208622399882582744059606376335285602101209468687677811493689474942744891506041932

comment: A root of $P_{18}$, with minimal polynomial $x^{19} - x^{17} - x^{16} - x^{15} - x^{14} - x^{13} - x^{12} - x^{11} - x^{10} - x^{9} - x^{8} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x - 1$.

36:

1.617956528253976570213724233669342650311982524814136763277345754456220908746937624048581615827138001

comment: A root of $Q_{19}$, with minimal polynomial $x^{21} - x^{20} - x^{19} + x^{2} - 1$.

37:

1.617986125385249151581179065219836276338293750679622676312918048191078288185448939191521385866321094

comment: A root of $P_{19}$, with minimal polynomial $x^{19} - x^{18} - x^{16} - x^{14} - x^{12} - x^{10} - x^{8} - x^{6} - x^{4} - x^{2} - 1$.

38:

1.617986128552861828746775573560971560626559554179903857177524556107337154595403717855874656309277088

comment: A root of $Q_{20}$, with minimal polynomial $x^{22} - x^{21} - x^{20} + x^{2} - 1$.

39:

1.618004413617124503253339884205705740949886990727485864190374404055016480918880509307231034414017485

comment: A root of $P_{20}$, with minimal polynomial $x^{21} - x^{19} - x^{18} - x^{17} - x^{16} - x^{15} - x^{14} - x^{13} - x^{12} - x^{11} - x^{10} - x^{9} - x^{8} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x - 1$.

40:

1.618004414826328965224816813048886874741852358452825536420120516315769101498118708366546047544102188

comment: A root of $Q_{21}$, with minimal polynomial $x^{23} - x^{22} - x^{21} + x^{2} - 1$.

41:

1.618015712751258280635900473733667949090056927598243987323992158078160732484712147251547557149689392

comment: A root of $P_{21}$, with minimal polynomial $x^{21} - x^{20} - x^{18} - x^{16} - x^{14} - x^{12} - x^{10} - x^{8} - x^{6} - x^{4} - x^{2} - 1$.

42:

1.618015713212954752257255964375369229374123216484787415640736393798376918798139605181967819732712222

comment: A root of $Q_{22}$, with minimal polynomial $x^{24} - x^{23} - x^{22} + x^{2} - 1$.

43:

1.618022694541460339603743229827892343352486377764735927967607280632891128415448447893953664301512573

comment: A root of $P_{22}$, with minimal polynomial $x^{23} - x^{21} - x^{20} - x^{19} - x^{18} - x^{17} - x^{16} - x^{15} - x^{14} - x^{13} - x^{12} - x^{11} - x^{10} - x^{9} - x^{8} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x - 1$.

44:

1.618022694717768294416502444185158998331096020868084111924521311236631268730313972376491207972498484

comment: A root of $Q_{23}$, with minimal polynomial $x^{25} - x^{24} - x^{23} + x^{2} - 1$.

45:

1.618027008938025718553681758942751295768660974229159521330823899915592183362260151838715294021339562

comment: A root of $P_{23}$, with minimal polynomial $x^{23} - x^{22} - x^{20} - x^{18} - x^{16} - x^{14} - x^{12} - x^{10} - x^{8} - x^{6} - x^{4} - x^{2} - 1$.

46:

1.618027009005358347427925536092720504146670868386797655664278670383537383085953832521182612767970098

comment: A root of $Q_{24}$, with minimal polynomial $x^{26} - x^{25} - x^{24} + x^{2} - 1$.

47:

1.618029675146055305698946786118197148936629694393732406286549986112999293588968023123008043461804993

comment: A root of $P_{24}$, with minimal polynomial $x^{25} - x^{23} - x^{22} - x^{21} - x^{20} - x^{19} - x^{18} - x^{17} - x^{16} - x^{15} - x^{14} - x^{13} - x^{12} - x^{11} - x^{10} - x^{9} - x^{8} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x - 1$.

48:

1.618029675171771354015094432833994648403067615748015040807374613270327489817157619418160494014043687

comment: A root of $Q_{25}$, with minimal polynomial $x^{27} - x^{26} - x^{25} + x^{2} - 1$.

49:

1.618031322858835939379993750912293610423937558487693101083392560992438305802819245773759782326311482

comment: A root of $P_{25}$, with minimal polynomial $x^{25} - x^{24} - x^{22} - x^{20} - x^{18} - x^{16} - x^{14} - x^{12} - x^{10} - x^{8} - x^{6} - x^{4} - x^{2} - 1$.

50:

1.618031322868657922029597723248317005726758817620183761258070106896742787251271592809502788045381653

comment: A root of $Q_{26}$, with minimal polynomial $x^{28} - x^{27} - x^{26} + x^{2} - 1$.

51:

1.618032341163605579648903926964992013213279130267502064682574041562690271509451899678934653848486982

comment: A root of $P_{26}$, with minimal polynomial $x^{27} - x^{25} - x^{24} - x^{23} - x^{22} - x^{21} - x^{20} - x^{19} - x^{18} - x^{17} - x^{16} - x^{15} - x^{14} - x^{13} - x^{12} - x^{11} - x^{10} - x^{9} - x^{8} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x - 1$.

52:

1.618032341167357077065889192222780195487817567190728533925877968326127861216819008444098812251153325

comment: A root of $Q_{27}$, with minimal polynomial $x^{29} - x^{28} - x^{27} + x^{2} - 1$.

53:

1.618032970495511780338687616599930589208996178245445628013970733235115263164927424052538696348938110

comment: A root of $P_{27}$, with minimal polynomial $x^{27} - x^{26} - x^{24} - x^{22} - x^{20} - x^{18} - x^{16} - x^{14} - x^{12} - x^{10} - x^{8} - x^{6} - x^{4} - x^{2} - 1$.

54:

1.618032970496944683958594372622324706609889729350103785351949963500471438131217303727853858919894804

comment: A root of $Q_{28}$, with minimal polynomial $x^{30} - x^{29} - x^{28} + x^{2} - 1$.

55:

1.618033359438026026238901165491703242086464985017968210215095423162683971400228321817036954102863048

comment: A root of $P_{28}$, with minimal polynomial $x^{29} - x^{27} - x^{26} - x^{25} - x^{24} - x^{23} - x^{22} - x^{21} - x^{20} - x^{19} - x^{18} - x^{17} - x^{16} - x^{15} - x^{14} - x^{13} - x^{12} - x^{11} - x^{10} - x^{9} - x^{8} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x - 1$.

56:

1.618033359438573336673250396246203626459581840869958339273054776166026984887408205988724420868884110

comment: A root of $Q_{29}$, with minimal polynomial $x^{31} - x^{30} - x^{29} + x^{2} - 1$.

57:

1.618033599815336574567100631932552880608568335391982437750809827236041127145973301722064897213120794

comment: A root of $P_{29}$, with minimal polynomial $x^{29} - x^{28} - x^{26} - x^{24} - x^{22} - x^{20} - x^{18} - x^{16} - x^{14} - x^{12} - x^{10} - x^{8} - x^{6} - x^{4} - x^{2} - 1$.

58:

1.618033599815545626086012322254917375874957691369417647049074584787004197809450663427506678804394252

comment: A root of $Q_{30}$, with minimal polynomial $x^{32} - x^{31} - x^{30} + x^{2} - 1$.

59:

1.618033748375738728063757479429073476875277793245441539038578830560588828285930877271489710587575927

comment: A root of $P_{30}$, with minimal polynomial $x^{31} - x^{29} - x^{28} - x^{27} - x^{26} - x^{25} - x^{24} - x^{23} - x^{22} - x^{21} - x^{20} - x^{19} - x^{18} - x^{17} - x^{16} - x^{15} - x^{14} - x^{13} - x^{12} - x^{11} - x^{10} - x^{9} - x^{8} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x - 1$.

60:

1.618033748375818578034781444715890397100131095987045983757988370269097379795341674587381629088467488

comment: A root of $Q_{31}$, with minimal polynomial $x^{33} - x^{32} - x^{31} + x^{2} - 1$.

61:

1.618033840190741707941604430175133749350573967093245095758996733010298160727195647031278557280865402

comment: A root of $P_{31}$, with minimal polynomial $x^{31} - x^{30} - x^{28} - x^{26} - x^{24} - x^{22} - x^{20} - x^{18} - x^{16} - x^{14} - x^{12} - x^{10} - x^{8} - x^{6} - x^{4} - x^{2} - 1$.

62:

1.618033840190772207768806359894125977128245225291216184439199781542680087108527631920216250435083937

comment: A root of $Q_{32}$, with minimal polynomial $x^{34} - x^{33} - x^{32} + x^{2} - 1$.

63:

1.618033896935385806005075010892848497468544878080385320807371408703642480600559510651605294301488646

comment: A root of $P_{32}$, with minimal polynomial $x^{33} - x^{31} - x^{30} - x^{29} - x^{28} - x^{27} - x^{26} - x^{25} - x^{24} - x^{23} - x^{22} - x^{21} - x^{20} - x^{19} - x^{18} - x^{17} - x^{16} - x^{15} - x^{14} - x^{13} - x^{12} - x^{11} - x^{10} - x^{9} - x^{8} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x - 1$.

64:

1.618033896935397455866315731334333523030485855870201521350548603175221775481807433193019461838877597

comment: A root of $Q_{33}$, with minimal polynomial $x^{35} - x^{34} - x^{33} + x^{2} - 1$.

65:

1.618033932005445856718849527159699357061247578236291468094541319022086440629526703792222680229254975

comment: A root of $P_{33}$, with minimal polynomial $x^{33} - x^{32} - x^{30} - x^{28} - x^{26} - x^{24} - x^{22} - x^{20} - x^{18} - x^{16} - x^{14} - x^{12} - x^{10} - x^{8} - x^{6} - x^{4} - x^{2} - 1$.

66:

1.618033932005450306561068039609331972320887882266570704364548928079413753979856803952415697269285063

comment: A root of $Q_{34}$, with minimal polynomial $x^{36} - x^{35} - x^{34} + x^{2} - 1$.

67:

1.618033953679911781204659054112901327706912927903258572681967179883618897198505908203606095952587995

comment: A root of $P_{34}$, with minimal polynomial $x^{35} - x^{33} - x^{32} - x^{31} - x^{30} - x^{29} - x^{28} - x^{27} - x^{26} - x^{25} - x^{24} - x^{23} - x^{22} - x^{21} - x^{20} - x^{19} - x^{18} - x^{17} - x^{16} - x^{15} - x^{14} - x^{13} - x^{12} - x^{11} - x^{10} - x^{9} - x^{8} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x - 1$.

68:

1.618033953679913480890993605302364766746072201143781966660388875616136647727956872540428361222662863

comment: A root of $Q_{35}$, with minimal polynomial $x^{37} - x^{36} - x^{35} + x^{2} - 1$.

69:

1.618033967075459268399127022246199518343981850399448564212910512812007121990987833866902842256579496

comment: A root of $P_{35}$, with minimal polynomial $x^{35} - x^{34} - x^{32} - x^{30} - x^{28} - x^{26} - x^{24} - x^{22} - x^{20} - x^{18} - x^{16} - x^{14} - x^{12} - x^{10} - x^{8} - x^{6} - x^{4} - x^{2} - 1$.

70:

1.618033967075459917621013327690153278861354496592865079073234665376206020469112492741119365891927508

comment: A root of $Q_{36}$, with minimal polynomial $x^{38} - x^{37} - x^{36} + x^{2} - 1$.

71:

1.618033975354359310516292063083806439836116958073550728097699983857318505921306092060792140507284612

comment: A root of $P_{36}$, with minimal polynomial $x^{37} - x^{35} - x^{34} - x^{33} - x^{32} - x^{31} - x^{30} - x^{29} - x^{28} - x^{27} - x^{26} - x^{25} - x^{24} - x^{23} - x^{22} - x^{21} - x^{20} - x^{19} - x^{18} - x^{17} - x^{16} - x^{15} - x^{14} - x^{13} - x^{12} - x^{11} - x^{10} - x^{9} - x^{8} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x - 1$.

72:

1.618033975354359558496859055597654471874124132146971613679444727538843318112721992952031807351259171

comment: A root of $Q_{37}$, with minimal polynomial $x^{39} - x^{38} - x^{37} + x^{2} - 1$.

73:

1.618033980470999507470063682523934060354736726182357616417617300460265088628911677060779065075629550

comment: A root of $P_{37}$, with minimal polynomial $x^{37} - x^{36} - x^{34} - x^{32} - x^{30} - x^{28} - x^{26} - x^{24} - x^{22} - x^{20} - x^{18} - x^{16} - x^{14} - x^{12} - x^{10} - x^{8} - x^{6} - x^{4} - x^{2} - 1$.

74:

1.618033980470999602190180765712365012153208665287157984545383021640954988152388716701594216151111797

comment: A root of $Q_{38}$, with minimal polynomial $x^{40} - x^{39} - x^{38} + x^{2} - 1$.

75:

1.618033983633256499378468705112305917526639513398821651309315804743089977306817600610105435980118153

comment: A root of $P_{38}$, with minimal polynomial $x^{39} - x^{37} - x^{36} - x^{35} - x^{34} - x^{33} - x^{32} - x^{31} - x^{30} - x^{29} - x^{28} - x^{27} - x^{26} - x^{25} - x^{24} - x^{23} - x^{22} - x^{21} - x^{20} - x^{19} - x^{18} - x^{17} - x^{16} - x^{15} - x^{14} - x^{13} - x^{12} - x^{11} - x^{10} - x^{9} - x^{8} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x - 1$.

76:

1.618033983633256535558326492090225455042348556575736721109974400639838819754475866391279579998441504

comment: A root of $Q_{39}$, with minimal polynomial $x^{41} - x^{40} - x^{39} + x^{2} - 1$.

77:

1.618033985587638582217684769585676076875498487327003805728840284453978706065673687543925706953304938

comment: A root of $P_{39}$, with minimal polynomial $x^{39} - x^{38} - x^{36} - x^{34} - x^{32} - x^{30} - x^{28} - x^{26} - x^{24} - x^{22} - x^{20} - x^{18} - x^{16} - x^{14} - x^{12} - x^{10} - x^{8} - x^{6} - x^{4} - x^{2} - 1$.

78:

1.618033985587638596037158910664826353779888507037442444226853378161231125074096695799307510089400812

comment: A root of $Q_{40}$, with minimal polynomial $x^{42} - x^{41} - x^{40} + x^{2} - 1$.

79:

1.618033986795513050281884574333095015899117245081282455987875605265561399710086506315603741611847492

comment: A root of $P_{40}$, with minimal polynomial $x^{41} - x^{39} - x^{38} - x^{37} - x^{36} - x^{35} - x^{34} - x^{33} - x^{32} - x^{31} - x^{30} - x^{29} - x^{28} - x^{27} - x^{26} - x^{25} - x^{24} - x^{23} - x^{22} - x^{21} - x^{20} - x^{19} - x^{18} - x^{17} - x^{16} - x^{15} - x^{14} - x^{13} - x^{12} - x^{11} - x^{10} - x^{9} - x^{8} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} - x^{2} - x - 1$.

Definition

Let $\varphi=(1+\sqrt5)/2$ be the golden ratio. For an integer $r\geq1$, $\theta_r$ is the $r$-th smallest Pisot-Vijayaraghavan number [5] in the interval $(1,\varphi)$.

Parameters

$r$

— rank ($r\geq1$)

Formulas

(1)

The set $\{\theta_r:r\geq1\}$ consists of the roots in $(1,\varphi)$ of $P_n(x)=x^n(x^2-x-1)+1$ and $Q_n(x)=x^n(x^2-x-1)+x^2-1$ for $n\geq2$, together with the root in $(1,\varphi)$ of $E(x)=x^6-2x^5+x^4-x^2+x-1$.

Comments

(2)

Dufresnoy and Pisot proved that $\varphi$ is the smallest limit point of the set of Pisot numbers and determined all Pisot numbers in $(1,\varphi)$ [1]. The Pisot numbers in $(1,\varphi)$ therefore have a well-defined increasing order and converge to $\varphi$. They are the roots in $(1,\varphi)$ of the polynomials in (1) [3] [4].

(3)

For every $n\geq2$, the root of $P_n$ in $(1,\varphi)$ is the growth rate, with respect to the three reflection generators, of the Coxeter triangle group $\Delta(2,n+1,\infty)$ [2].

(4)

Each number is an algebraic unit. Its conjugates other than itself lie inside the unit disc, so it is the Mahler measure of its minimal polynomial.

Programs

(P1)

Sage

R.<x> = QQ[]
P = lambda n: x^n*(x^2 - x - 1) + 1
Q = lambda n: x^n*(x^2 - x - 1) + x^2 - 1
E = x^6 - 2*x^5 + x^4 - x^2 + x - 1
IR = RealIntervalField(400)
phi = (IR(1) + IR(5).sqrt())/2
def pisot_root(f):
    out = []
    for g, e in f.factor():
        out += [(g, root) for root, m in g.roots(IR)
                if root.lower() > 1 and root.upper() < phi.lower()]
    return out[0]
polys = [P(n) for n in range(2, 41)] + [Q(n) for n in range(2, 41)] + [E]
sorted((pisot_root(f) for f in polys), key=lambda t: t[1].lower())[:10]

References

[1]

J. Dufresnoy and Ch. Pisot, Etude de certaines fonctions meromorphes bornees sur le cercle unite. Application a un ensemble ferme d'entiers algebriques, Annales Scientifiques de l'Ecole Normale Superieure 72 (1955), 69-92.

[2]

W. J. Floyd, Growth of planar Coxeter groups, P.V. numbers, and Salem numbers, Math. Ann. 293 (1992), no. 3, 475-483. (zbMATH) (MR)

[3]

M.-J. Bertin, A. Decomps-Guilloux, M. Grandet-Hugot, M. Pathiaux-Delefosse and J.-P. Schreiber, Pisot and Salem Numbers, Birkhauser, 1992. (doi)

[4]

James McKee and Chris Smyth, Salem numbers, Pisot numbers, Mahler measure and graphs, Experimental Mathematics 14 (2005), 211-229. (arXiv) (doi)

Links

[5]

Wikipedia: Pisot-Vijayaraghavan number

Similar tables

Growth rates of hyperbolic Coxeter triangle groups — holds the same numbers as growth rates of the Coxeter groups $\Delta(2,n+1,\infty)$ for the overlapping roots of $P_n$

Minimal polynomials of the Pisot numbers less than the golden ratio — holds the minimal polynomial of each $\theta_r$ here, over the same range

Salem numbers less than $1.3$ — stores Salem numbers, the companion class of algebraic integers in Lehmer's problem

Golden ratio — holds the smallest accumulation point of the set of Pisot numbers

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds the root of $E$ and the roots of $P_n$ and $Q_n$ for $2\leq n\leq40$, which are $\theta_1$ to $\theta_{79}$; the cutoff keeps the largest root more than $1.9\cdot10^{-9}$ below the accumulation point $\varphi$, while by $n=72$ the roots are within $10^{-15}$ of $\varphi$ and close to the resolution of double-precision output)

How they were obtained:

The generator uses the Dufresnoy-Pisot classification as recorded in [3] and [4]. It forms the exact integer polynomials in (1), factors them over $\mathbb Q$, selects the unique irreducible factor with a real root in $(1,\varphi)$, and isolates that root in interval arithmetic before writing $100$ digits. It verifies that the first ten minimal polynomials agree with the source table quoted by Wikipedia's small-Pisot-numbers section, including the plastic and supergolden ratios.

The roots from $P_n$ and $Q_n$ used here increase with $n$: at a root of $P_n$, $P_{n+1}(x)=1-x<0$, and at a root of $Q_n$, $Q_{n+1}(x)=-(x-1)^2(x+1)<0$, while both next polynomials are positive at $\varphi$. The computed ordering has $P_{41}$ and $Q_{41}$ larger than $\theta_{79}$, so the first $79$ ranks are accounted for.