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  Types de Nombres

 

Débutants

Premier

NOMBRES PREMIERS

 

Glossaire

Premier

 

 

INDEX

 

Premiers

Types de premiers

Cullen

Woodall

Carol

 

Sommaire de cette page

>>> Nombres premiers de Woodall

>>> Valeurs

>>> Liste

 

 

 

 

 

 

Nombres premiers de Woodall

Famille

Nombre / Diviseurs / Multiplicatif / Premiers

 

… / Types de nombres premiers et cousins / Cullen

Noms

Premier de Woodall

Premier de Riesel

Premier de Cullen du second type (Cullen primes of the second kind)

Définitions

NOMBRES PREMIERS DE WOODALL

 

Nombre premier de la forme Wn = n . 2n – 1

Exemples

n = 1 => 1 x 21 – 1 = 1        non premier

n = 2 => 2 x 22 – 1 = 7        Premier

n = 3 => 3 x 23 – 1 = 23      Premier

n = 4 => 4 x 24 – 1 = 63      non premier

n = 5 => 5 x 25 – 1 = 159    non premier

n = 6 => 6 x 26 – 1 = 383    Premier

Les suivants

895, 2047, 4607, 10239, 22527, 49151, 106495, 229375, 491519, 1048575, 2228223, 4718591, 9961471, 20971519, 44040191, 92274687, 192937983, 402653183, 838860799, 1744830463, 3623878655, 7516192767, …

Généralisation

NOMBRES PREMIERS DE WOODALL généralisés d'ordre b

 

Nombre premier de la forme W'n = n . bn – 1

b est l'ordre,

n est le générateur.

Propriétés

*    Bien que rares et vite très grands, on conjecture qu'ils sont en nombre infini.

*    Presque tous les nombres de Woodall sont composés. Démontré par Christopher Hooley en 1976; puis par Hiromi Suyama pour n.2n+a + b).

*    Aucun premier de Woodall et de Cullen n'est Fibonacci sauf cas triviaux: F4 = 3 = 1x21 + 1 pour Cullen et F1 = F2 = 1 = 1x21 – 1 pour Woodall (Luca et Stanica – 2003).

Divisibilité

Si p est un nombre premier alors, l'un de ces nombres de Woodall est divisible par p.

W(p + 1)/2 ou W(3p – 1)/2

Selon que le symbole de Jacobi est égal à + 1 ou  à -1, respectivement.

Historique

Étudiés par Cunningham puis par Woodall en 1917. Cullen avait étudié des nombres similaires avant eux.

Anglais

Woodall prime numbers.

 

Un beau spécimen

2521  – 1 = 512  2512 – 1

= 6,8648 … 10156

Premier de Mersenne et de Woodall

(Trouvé par Dobb, cité par Caldwell)

 

= 6864797 6601306097 1498190079 9081393217 2694353001 4330540939 4463459185 5431833976 5605212255 9640661454 5549772963 1139148085 8037121987 9997166438 1257402829 1115057151

 

 

 

Valeurs des nombres premiers de Woodall

 

Inférieurs à 1000 et d'ordre 2 à 10 (ordre indiqué en indice)

 

 

Valeurs des nombres premiers de Woodall

 

Ordre (valeur de b)

2

3

4

5

6

7

8

9

10

Premier Woodall jusqu'à p = 1000  (valeur de n, nombre de Woodall)

  2, 7

  3, 23

  6, 383

1, 2

2, 17

1, 3

2, 31

3, 191

 

1, 5

2, 71

3, 647

2, 97

1, 7

2, 127

 

2, 199

Générateur jusqu'à n = 1000

 

2

  3

 

6

 30

 75

 81

115

123

249

362

384

462

512

751

822

 

1

2

 

 

6

10

18

40

46

86

118

170

 

1

2

3

5

8

14

23

63

107

132

428

530

 

 

 

 

 

8

14

42

384

564

 

1

2

3

 

 

19

20

24

34

77

107

114

122

165

530

 

 

2

 

 

 

18

68

84

 

1

2

7

 

 

12

25

44

219

252

507

 

 

 

 

 

 

10

58

264

 

2

3

8

 

11

15

39

60

72

77

117

183

252

396

 

 

 

Liste des premiers de Woodall  (ordre 2)

 

Il y en 15 jusqu'à n = 1000. Voici la liste avec n puis p.

 

  2, 7

  3, 23

  6, 383

 30, 32212254719

 75, 2833419889721787128217599

 81, 195845982777569926302400511

115, 4776913109852041418248056622882488319

123, 1307960347852357218937346147315859062783

249, 225251798594466661409915431774713195745814267044878909733007331390393510002687

362, 340068965985651051398396790415096001563998833081924717278415732677863866836595
3026135436149661145081990241845247

384, 151303703794154800175151513984551477011506198798587315204921446672303571602549
28874783078241875807606069745148277817343

462, 550173885439296116217917602601792462637272774471747082403919416559952447255985
9148008657148801167643264160317146644057379520281844448806043647

512, 686479766013060971498190079908139321726943530014330540939446345918554318339765
60521225596406614545549772963113914808580371219879997166438125740282911150571
51

751, 889542455534234932172304989498397383879029387553893398258535828062106525777793
15310248541643548088450684836254390368345476902512598436401975605754360651171
53492322441037431938692087017130511719236545657306526703124393676660277247

822, 22989432637682048935578359759258512929075458593285426151563351225878608019921
96017478693717432406691855755226228322047841909591752179132387477130020133406
68438101393370692503399055767938825396035873270378579048763918114404929084899
72485276368673701887

 

Quelques valeurs suivantes de n (générateur)


 5 312     7 755     9 531    12 379    15 822    8 885    etc.

 

Le plus grand premier de Woodall connus en 2015 a plus de 1,2 millions de chiffres.

 

 

           

Liste des premiers généralisés de Woodall 

 

Ordre 3

Il y en 10 jusqu'à n = 1000. Voici la liste avec n puis p.

 

  1, 2

  2, 17

  6, 4373

10, 590489

18, 6973568801

40, 486306618362277152039

46, 407695153504015050412733

86, 9266726751303003316378520780678994459797093

118, 23560801709989209203195024431348154965368236005496270061701

170, 219311913288917453262193326093661967138685096079648824587719795028581573749
184854329

 

Ordre 4

Il y en 12 jusqu'à n = 1000.

 

1, 3

2, 31

3, 191

5, 5119

8, 524287

14, 3758096383

23, 1618481116086271

63, 5359447279004780799548150067050349330431

107, 2817103802133904744169307240538184064530443801964688726052818649087

132, 3912846279507388875753310725413581782177699321821325940021350678795
400476792717311

428, 205645900974618348699369951595389563945051718786508000876681020111
77000376326599494409546015951243718420359169350745203206465226273
13510474261699371546802821438081998636100033984049237601634711342
68885802807188735619416647112001032130896303599267059269116100607

530, 65474361726608049149671956641251326132974258041378981157933178561
11943357607271353316589560654399168850656205327631244692175209400
29727433712872936820015931752955535074341867206832648335628690556
98899456537203910118325174691825641032467516170687600622942706269
25668204512738631962412496463546971410057575656316229417697279

 

Ordre 5

Il y en 5 jusqu'à n = 1000.

 

8, 3124999

14, 85449218749

42, 9549694368615746498107910156249

384, 97456966552920930173722542943892872198026963828522070333672716335
5959051989490997990044452757421165912191202966176344005095195395
4983526538677912413476530349204671358985069056829409871366056218
2692588563741979304373099413007913716100460987945552915334701538
08593749999999

564, 9340315570753691350383239728574424197216807698206663098370619779
7829884547738881738965291705275390678939574234635835586087466234
5440426913903796838110057568992468343542288677229755732930598252
8164733162205681899887141069826972201752152960825656563829468450
4358792620980493564373276957425592438026630771013929306850357324
5342169166553896722334754253196835785399976259668619604781270027
1606445312499

 

Ordre 6

Il y en 13 jusqu'à n = 1000.

 

1, 5

2, 71

3, 647

19, 11577835060199423

20, 73123168801259519

24, 113721152119718805503

34, 9741401198574394682495729663

77, 63699643930293116661668059033734770664712983894089510286262271

107, 1956895203412839586109189026910591392333778720564040915647010915560443
6042237347889151

114, 5836431531358293412241594494938853978364088739332612222890843722714900
079465906363846098943

122, 1049089106598960302143210976722604049100285287003817497356394157158705
4146038678504847043662446591

165, 4096757500580606540655605983026061505881578187370285973482045398232347
6114410256378188900530715106962760862441613848446833304535039

530, 139456439503569517710652068371761530269197876967172036106169051766289
686685974970690313784062774289654072635078702610380922238610678966737
696066303247783591807149836652210267852260476872958329713659745169066
104234978494943491128982062136257863189477894715492046095182421794609
960898748257869444908990828406480921386348222549956234964368083307511
700961094577068194579374775177688570683776831075580177693258340420812
79

 

Ordre 7

Il y en 4 jusqu'à n = 1000.

 

2, 97

18, 29311444762388081

68, 199147855295327623090224677392931283231127797842219031254467

84, 8175506106494468513838372140330918838855718324473121098267078982890633683

 

Ordre 8

Il y en 9 jusqu'à n = 1000.

 

1, 7

2, 127

7, 14680063

12, 824633720831

25, 944473296573929042739199

44, 239558786312340678278215723631964820865023

219, 1309637994426218254018041061898143400134727490715562311413718268536345410
428862919664985578770929204004811788512942063879508231892475865886382270
48565199982720081832664746953810660484461626160178003967

252, 9551624981928189737732979274720474705193732331870301416187527187074336915
941576546762240210566867979578195999406177501170782181912034502603612314
289804141774402408543878275786100264940897529494530426655195716555791625
9132607823871

507, 372933471816700293704259985124620134047372826143940520733445814454982146
465162421490242342291999648187063148347293157479079054384841627157824583
064227484969570249378377222897021114048310627600200244351309404182363648
026840197031173247352072419571087113614663745283948839765317116632091126
946911966451243981255208538322554661599272347980443888608048660595827271
671585201331378125613106731368734985723085350545417898197828346955368225
88057599643687027020675416063

 

Ordre 9

Il y en 3 jusqu'à n = 1000.

 

10, 34867844009

58, 1286748115988111237085980618661198670425007239471547705817

264, 219597008140207558063412054892183314028390644062528823147577868513896629
87260802602450659767802914996348735978778998233115810430847239451626767
70848755180798099173095842520786759425701803065022209045539359643239099
84843464883549278334340140224458535549703

 

Ordre 10

Il y en 13 jusqu'à n = 1000.

 

2, 199

3, 2999

8, 799999999

11, 1099999999999

15, 14999999999999999

39, 38999999999999999999999999999999999999999

60, 59999999999999999999999999999999999999999999999999999999999999

72, 71999999999999999999999999999999999999999999999999999999999999999999999999

77, 76999999999999999999999999999999999999999999999999999999999999999999999999
99999

117, 1169999999999999999999999999999999999999999999999999999999999999999999999
99999999999999999999999999999999999999999999999

183, 1829999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999
99999999999999999999999999999999999999999

252, 2519999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999
99999999999999999999999999999999999999

396, 395999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999

 

 

 

 

 

Suite

*         Types de nombres premiersIndex

*         Nombres premiers records

Voir

*         Place de ces nombres parmi les autres premiers

*           Records

DicoNombre

*         Nombre 2521 – 1

Site

*         OEIS A003261

*         Woodall prime – Chris Caldwell

*         Woodall numbers explained

Cette page

http://villemin.gerard.free.fr/aNombre/TYPMULTI/PremWood.htm