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Tabelas

Nesta �ltima se��o apresentaremos algumas tabelas indicando os maiores primos conhecidos no momento da conclus�o do livro (14 de junho de 1999). Lembramos (veja a introdu��o) que existe um n�mero de Mersenne, maior do que todos estes, cuja primalidade estava sendo checada no momento de imprimir este livro.

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Os dez maiores primos conhecidos

Primo	N^o de d�gitos	Descobridores

2^3021377-1	909526	Clarkson, Woltman, Kurowski & al. (GIMPS)
2^2976221-1	895932	Spence, Woltman & al. (GIMPS)
2^1398269-1	420921	Armengaud, Woltman & al. (GIMPS)
2^1257787-1	378632	Slowinski, Gage
2^859433-1	258716	Slowinski, Gage
2^756839-1	227832	Slowinski, Gage
302627325 2^530101+1	159585	Nash, Dunaieff, Burrowes, Jobling, Gallot
481899 2^481899+1	145072	Morii, Gallot
361275 2^361275+1	108761	Smith, Gallot
302442855 2^336211+1	101219	Nash, Dunaieff, Burrowes, Jobling & Gallot

Lembramos que quando p e p+2 s�o ambos primos, dizemos que eles s�o primos g�meos.

Os dez maiores pares de primos g�meos conhecidos

Primo	N^o de d�gitos	Descobridores

361700055 2^39020 1	11755	Henri Lifchitz
835335 2^39014 1	11751	Ballinger & Gallot
242206083 2^38880 1	11713	J�rai & Indlekofer
40883037 2^23456 1	7069	Lifchitz & Gallot
843753 2^22222 1	6696	Rivera & Gallot
7485 2^20023 1	6032	Buddenhagen & Gallot
8182815 2^17838 1	5377	Smith & Gallot
570918348 10^5120 1	5129	Harvey Dubner
697053813 2^16352 1	4932	J�rai & Indlekofer
37442007 2^15440 1	4656	Hanson & Gallot

Seja $n\char93$ , chamado o primorial de n, o produto de todos os n�meros primos menores ou iguais a n. Usamos tamb�m a nota��o $n!!\ldots !!$ , com k pontos de exclama��o, para o produto $n(n-k)(n-2k)\ldots$ dos inteiros positivos menores ou iguais a n e c�ngruos a n m�dulo k. Um primo da forma $n\char93 \pm 1$ � chamado primorial e um primo da forma $n!!\ldots !! \pm 1$ � chamado multifatorial.

Os dez maiores primos multifatoriais e primoriais conhecidos

Primo	N^o de d�gitos	Descobridores

6917!-1	23560	Caldwell & Gallot
6380!+1	21507	Caldwell & Gallot
42209#+1	18241	Caldwell & PrimeForm
14614!!!!+1	13632	Charles F. Kerchner III
10830!!!+1	13000	Charles F. Kerchner III
3610!-1	11277	Chris Caldwell
3507!-1	10912	Chris Caldwell
24029#+1	10387	Chris Caldwell
23801#+1	10273	Chris Caldwell
11915!!!!!+1	8681	Charles F. Kerchner III

Lembramos que p � dito um primo de Sophie Germain se 2p+1tamb�m � primo e que M_p � composto para estes valores de pse $p\equiv 3 \pmod 4$ . Este nome � usado porque Sophie Germain provou o primeiro caso do �ltimo teorema de Fermat (recentemente demonstrado completamente por Wiles) para primos p desta forma.

Os dez maiores primos de Sophie Germain conhecidos

Primo N^o de d�gitos Descobridores

18458709.2^32611-1 9825 Kerchner & Gallot

14516877.2^24176-1 7285 Kerchner & Gallot

72021.2^23630-1 7119 Yves Gallot

2375063906985.2^19380-1 5847 J�rai & Indlekofer

276311.2^19003+1 5726 Ballinger & Gallot

92305.2^16998+1 5122 Kerchner & Gallot

8069496435.10^5072-1 5082 Harvey Dubner

470943129.2^16352-1 4932 J�rai & Indlekofer

157324389.2^16352-1 4931 J�rai & Indlekofer

5415312903.10^4526-1 4536 Harvey Dubner

O maior primo conhecido ao longo da hist�ria

Primo N^o de d�gitos Data Descobridores

2^17-1 6 1588 Cataldi

2^19-1 6 1588 Cataldi

2^31-1 10 1772 Euler

999999000001 12 1851 Loof

(2^59-1)/179951 13 1867 Landry

(2^53+1)/(3 107) 14 1867 Landry

2^127-1 39 1876 Lucas

(2^148+1)/17 44 1951 Ferrier

180(2^127-1)^2+1 79 1951 Miller & Wheeler

2^521-1 157 1952 Robinson

2^607-1 183 1952 Robinson

2^1279-1 386 1952 Robinson

2^2203-1 664 1952 Robinson

2^2281-1 687 1952 Robinson

2^3217-1 969 1957 Riesel

2^4423-1 1332 1961 Hurwitz

2^9689-1 2917 1963 Gillies

2^9941-1 2993 1963 Gillies

2^11213-1 3376 1963 Gillies

2^19937-1 6002 1971 Tuckerman

2^21701-1 6533 1978 Noll & Nickel

2^23209-1 6987 1979 Noll

2^44497-1 13395 1979 Nelson & Slowinski

2^86243-1 25962 1982 Slowinski

2^132049-1 39751 1983 Slowinski

2^216091-1 65050 1985 Slowinski

91581 2^216193-1 65087 1989 Amdahl Six

2^756839-1 227832 1992 Slowinski & Gage

2^859433-1 258716 1994 Slowinski & Gage

2^1257787-1 378632 1996 Slowinski & Gage

2^1398269-1 420921 1996 Armengaud, Woltman, et. al. [GIMPS]

2^2976221-1 895932 1997 Spence, Woltman, et. al. [GIMPS]

2^3021377-1 909526 1998 Clarkson, Woltman, Kurowski, et. al. [GIMPS, PrimeNet]

Os cem maiores primos conhecidos

Primo N^o de d�gitos Data

1 2^3021377-1 909526 1998

2 2^2976221-1 895932 1997

3 2^1398269-1 420921 1996

4 2^1257787-1 378632 1996

5 2^859433-1 258716 1994

6 2^756839-1 227832 1992

7 302627325 2^530101+1 159585 1999

8 481899 2^481899+1 145072 1998

9 361275 2^361275+1 108761 1998

10 302442855 2^336211+1 101219 1998

11 9 2^304607+1 91697 1998

12 3 2^303093+1 91241 1998

13 7 2^283034+1 85203 1998

14 27253 2^272347-1 81990 1998

15 67234^16384+1 79096 1999

16 262419 2^262419+1 79002 1998

17 9183 2^262112+1 78908 1997

18 111113277 2^250132+1 75306 1998

19 22695 2^247131+1 74399 1999

20 217807 2^243537-1 73318 1999

21 5 2^240937+1 72530 1997

22 982451707 2^239848+1 72211 1998

23 25229 2^238652-1 71846 1998

24 73 2^227334+1 68437 1999

25 127 2^220417-1 66355 1999

26 29 2^219317+1 66023 1999

27 391581 2^216193-1 65087 1989

28 2^216091-1 65050 1985

29 3 2^213321+1 64217 1997

30 5 2^209787+1 63153 1997

31 7 2^207084+1 62340 1998

32 132599 2^206032-1 62027 1999

33 331139 2^201240-1 60585 1999

34 281143 2^187639-1 56491 1999

35 81 2^185745+1 55917 1999

36 15 2^184290+1 55478 1998

37 60541 2^176340+1 53089 1997

38 39781 2^176088+1 53013 1997

39 73 2^171854+1 51736 1998

40 127 2^170393-1 51296 1999

41 159821 2^168770-1 50811 1999

42 48833 2^167897+1 50547 1999

43 74269 2^167546+1 50442 1999

44 2 3^105106+1 50149 1999

45 285 2^165957+1 49961 1998

46 111253 2^165379-1 49790 1999

47 21 2^164901+1 49642 1999

48 1002774^8192+1 49162 1999

49 27923 2^158625+1 47756 1997

50 3 2^157169+1 47314 1995

51 325859 2^156148-1 47011 1999

52 285 2^155637+1 46854 1998

53 111763 2^155551-1 46831 1999

54 291 2^154544+1 46525 1999

55 151023 2^151023-1 45468 1998

56 19 2^149146+1 44899 1998

57 9 2^149143+1 44898 1995

58 185767 2^149009-1 44862 1999

59 256267 2^148941-1 44842 1999

60 165 2^147953+1 44541 1999

61 29 2^147316-1 44348 1999

62 9 2^147073+1 44275 1995

63 9 2^145247+1 43725 1995

64 29 2^144937+1 43632 1999

65 178747 2^144789-1 43592 1999

66 231 2^143949+1 43336 1998

67 165 2^143437+1 43182 1998

68 180924^8192+1 43070 1999

69 143018 2^143018-1 43058 1998

70 333 2^142307-1 42842 1998

71 190229 2^141576-1 42624 1999

72 63 2^141497+1 42597 1999

73 203 2^141477+1 42592 1999

74 285 2^141253+1 42524 1998

75 150152^8192+1 42407 1999

76 288759 2^140001+1 42150 1999

77 165 2^139459+1 41984 1998

78 126308^8192+1 41791 1999

79 81 2^138239+1 41616 1998

80 122463 2^137752+1 41473 1998

81 111850^8192+1 41359 1999

82 245 2^136993+1 41242 1999

83 130297 2^136645-1 41140 1999

84 70175 2^135753+1 40871 1998

85 438523 2^135415-1 40770 1999

86 203 2^135125+1 40679 1999

87 105 2^133443+1 40173 1998

88 71852^8192+1 39784 1999

89 2^132049-1 39751 1983

90 63 2^131325+1 39535 1999

91 2 3^82780+1 39497 1999

92 10038165 2^131040+1 39454 1997

93 5581 2^131000+1 39439 1999

94 577294575 2^130639+1 39336 1999

95 195 2^130388+1 39253 1998

96 63 2^130221+1 39203 1999

97 91 2^130140+1 39179 1999

98 577294575 2^129917+1 39118 1999

99 65 2^129925+1 39114 1999

100 113 2^129845+1 39090 1999

Obs: um primo � dito de Cullen se � da forma $n \cdot 2^n + 1$ , de Woodall se � da forma $n \cdot 2^n - 1$ e Fermat generalizado se � da forma a^2ⁿ + 1.

Next: Refer�ncias Up: Aspectos computacionais Previous: A complexidade das opera��es
Nicolau C. Saldanha
1999-08-09

Primo	N^o de d�gitos	Data	Descobridores

2^17-1	6	1588	Cataldi
2^19-1	6	1588	Cataldi
2^31-1	10	1772	Euler
999999000001	12	1851	Loof
(2^59-1)/179951	13	1867	Landry
(2^53+1)/(3 107)	14	1867	Landry
2^127-1	39	1876	Lucas
(2^148+1)/17	44	1951	Ferrier
180(2^127-1)^2+1	79	1951	Miller & Wheeler
2^521-1	157	1952	Robinson
2^607-1	183	1952	Robinson
2^1279-1	386	1952	Robinson
2^2203-1	664	1952	Robinson
2^2281-1	687	1952	Robinson
2^3217-1	969	1957	Riesel
2^4423-1	1332	1961	Hurwitz
2^9689-1	2917	1963	Gillies
2^9941-1	2993	1963	Gillies
2^11213-1	3376	1963	Gillies
2^19937-1	6002	1971	Tuckerman
2^21701-1	6533	1978	Noll & Nickel
2^23209-1	6987	1979	Noll
2^44497-1	13395	1979	Nelson & Slowinski
2^86243-1	25962	1982	Slowinski
2^132049-1	39751	1983	Slowinski
2^216091-1	65050	1985	Slowinski
91581 2^216193-1	65087	1989	Amdahl Six
2^756839-1	227832	1992	Slowinski & Gage
2^859433-1	258716	1994	Slowinski & Gage
2^1257787-1	378632	1996	Slowinski & Gage
2^1398269-1	420921	1996	Armengaud, Woltman, et. al. [GIMPS]
2^2976221-1	895932	1997	Spence, Woltman, et. al. [GIMPS]
2^3021377-1	909526	1998	Clarkson, Woltman, Kurowski, et. al. [GIMPS, PrimeNet]