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	Types de Nombres
Débutants Premier	NOMBRES PREMIERS				Glossaire Premier
INDEX Premiers	Types de premiers	Cullen	Woodall	Carol
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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 ^.2^n– 1
Exemples	n = 1 => 1 x 2^1 – 1 = 1 non premier n = 2 => 2 x 2^2 – 1 = 7 Premier n = 3 => 3 x 2^3 – 1 = 23 Premier n = 4 => 4 x 2^4 – 1 = 63 non premier n = 5 => 5 x 2^5 – 1 = 159 non premier n = 6 => 6 x 2^6 – 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 ^.b^n– 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.2^n+a + b). Aucun premier de Woodall et de Cullen n'est Fibonacci sauf cas triviaux: F4 = 3 = 1x2^1 + 1 pour Cullen et F1 = F2 = 1 = 1x2^1 – 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 Liste des valeurs de n (générateur) 2, 3, 6, 30, 75, 81, 115, 123, 249, 362, 384, 462, 512, 751, 822, 5312, 7755, 9531, 12379, 15822, 18885, 22971, 23005, 98726, 143018, 151023, 667071, 1195203, 1268979, 1467763, 2013992, 2367906, 3752948, 17016602, …. Le plus grand premier de Woodall connus en 2015 a plus de 1,2 millions de chiffres. En novembre 2021, le plus grand Woodall généralisé connu est: 2740879 × 32^2740879 − 1

Merci à Florina Petra

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 premiers – Index Nombres premiers records
Voir	Place de ces nombres parmi les autres premiers Records
DicoNombre	Nombre 2^521 – 1
Site	Nombre de Woodall – Wikipédia OEIS A003261 - Woodall (or Riesel) numbers: n*2^n - 1 OESI A002234 – Numbers k such that the Woodall number k*2^k - 1 is prime Woodall prime – Chris Caldwell Woodall numbers explained Woodall number – Wikipedia
Cette page	http://villemin.gerard.free.fr/aNombre/TYPMULTI/PremWood.htm