Indicamos aqui alguns dados sobre números primos “grandes”, obtidos por consulta a
“The Prime Pages”. O gráfico abaixo indica o (logaritmo do) número de dígitos do
maior primo conhecido em função do ano da descoberta, desde 1951, que marca o
início da “era do computador digital”. O recorde em Abril de 2005 é um primo de
Mersenne com quase 8.000.000 de dígitos, descoberto no passado mês de Fevereiro.
A tabela seguinte indica os detalhes dos sucessivos recordes desde 1951 (a versão
mais recente está em Maior Primo por Ano desde 1951). Os números de Mersenne
são Mn = 2n –1. Incluímos ainda dados sobre “primos gémeos”.
Logaritmo do Número de Dígitos do Maior Primo Conhecido
(Abril 2005)
8,00
7,00
6,00
5,00
4,00
3,00
2,00
1,00
0,00
1950
1960
1970
1980
1990
Ano
Number
Digits
Year Machine
Prover
180(M127)2+1
79
1951
EDSAC1
Miller & Wheeler
M521
157
1952
SWAC
Robinson (Jan 30)
M607
183
1952
SWAC
Robinson (Jan 30)
M1279
386
1952
SWAC
Robinson (June 25)
M2203
664
1952
SWAC
Robinson (Oct 7)
M2281
687
1952
SWAC
Robinson (Oct 9)
M3217
969
1957
BESK
Riesel
M4423
1,332
1961
IBM7090
Hurwitz
M9689
2,917
1963
ILLIAC 2
Gillies
M9941
2,993
1963
ILLIAC 2
Gillies
M11213
3,376
1963
ILLIAC 2
Gillies
M19937
6,002
1971
IBM360/91
Tuckerman
2000
M21701
6,533
1978
CDC Cyber
174
Noll & Nickel
M23209
6,987
1979
CDC Cyber
174
Noll
M44497
13,395
1979
Cray 1
Nelson & Slowinski
M86243
25,962
1982
Cray 1
Slowinski
M132049
39,751
1983
Cray X-MP
Slowinski
M216091
65,050
1985
Cray X-MP/24 Slowinski
65,087
1989
Amdahl 1200
Amdahl Six
M756839
227,832
1992
Cray-2
Slowinski & Gage
M859433
258,716
1994
Cray C90
Slowinski & Gage
M1257787
378,632
1996
Cray T94
Slowinski & Gage
M1398269
420,921
1996
Pentium (90
Mhz)
Armengaud, Woltman, et. al.
[GIMPS]
M2976221
895,932
1997
Pentium (100
Mhz)
Spence, Woltman, et. al.
[GIMPS]
M3021377
909,526
1998
Pentium (200
Mhz)
Clarkson, Woltman,
Kurowski, et. al. [GIMPS,
PrimeNet]
Hajratwala, Woltman,
Kurowski, et. al. [GIMPS,
PrimeNet]
216193
391581*2
-1
M6972593
2,098,960
1999
Pentium (350
Mhz)
M13466917
4,053,946
2001
AMD T-Bird
(800 Mhz)
Cameron, Woltman,
Kurowski, et. al. [GIMPS,
PrimeNet]
M20996011
6,320,430
2003
Pentium (2
GHz)
Shafer, Woltman, Kurowski,
et. al. [GIMPS, PrimeNet]
M24036583
7,235,733
2004
Pentium 4
(2.4GHz)
Findley, GIMPS et. al.
M25964951
7,816,230
2005
Pentium 4
(2.4GHz)
Nowak, GIMPS et. al.
All of the Mersenne records were found using the Lucas-Lehmer test and the other two were
found using Proth's Theorem (or similar results). The Amdahl Six is J. Brown, C Noll, B
Parady, G Smith, J Smith and S Zarantonello
Definitions and Notes
Twin primes are pairs of primes which differ by two. The first twin primes are {3,5}, {5,7},
{11,13} and {17,19}. It has been conjectured (but never proven) that there are infinitely many
twin primes. If the probability of a random integer n and the integer n+2 being prime were
statistically independent events, then it would follow from the prime number theorem that
there are about n/(log n)2 twin primes less than or equal to n. These probabilities are not
independent, so Hardy and Littlewood conjectured that the correct estimate should be the
following.
Here the infinite product is the twin prime constant (estimated by Wrench and others to be
approximately 0.6601618158...), and we introduce an integral to improve the quality of the
estimate. This estimate works quite well! For example:
The number of twin primes
less than N
N
actual
estimate
106
8169
8248
108
440312
440368
1010 27412679 27411417
There is a longer table by Kutnib and Richstein available online.
In 1919 Brun showed that the sum of the reciprocals of the twin primes converges to a sum
now called Brun's Constant. (Recall that the sum of the reciprocals of all primes diverges.) By
calculating the twin primes up to 1014 (and discovering the infamous pentium bug along the
way), Thomas Nicely heuristically estimates Brun's constant to be 1.902160578.
As an exercise you might want to prove the following version of Wilson's theorem .
Theorem: (Clement 1949)
The integers n, n+2, form a pair of twin primes if and only if
4[(n-1)!+1] = -n (mod n(n+2)).
Nice--too bad it is of virtually no practical value!
Record Primes of this Type
rank
1
2
3
prime
digits who when comment
33218925 · 2169690-1 51090 g259 2002 Twin (p)
60194061 · 2114689-1 34533 g294 2002 Twin (p)
1765199373 · 2107520-1 32376 g182 2002 Twin (p)
318032361 · 2107001-1 32220 p100 2001 Twin (p)
1807318575 · 298305-1 29603 g216 2001 Twin (p)
4
5
7
665551035 · 280025-1 24099 g216 2000 Twin (p)
1940734185 · 266445-1 20012 g336 2004 Twin (p)
8
781134345 · 266445-1 20011 p53 2001 Twin (p)
9
1693965 · 266443-1 20008 g183 2000 Twin (p)
10
83475759 · 264955-1 19562 g144 2000 Twin (p)
37831341 · 261777-1 18605 g277 2003 Twin (p)
6
11
291889803 · 260090-1 18098 g191 2001 Twin (p)
13 4648619711505 · 260000-1 18075 IJW 2000 Twin (p)
12
14 2409110779845 · 260000-1 18075 IJW 2000 Twin (p)
15
488162409 · 253333-1 16064 p135 2004 Twin (p)
16 2230907354445 · 248000-1 14462 IJW 1999 Twin (p)
17
871892617365 · 248000-1 14462 IJW 1999 Twin (p)
18
19
160675905 · 242004-1 12653 g350 2003 Twin (p)
1186468455 · 241537-1 12513 g294 2003 Twin (p)
20
1902519837 · 241536-1 12513 g294 2003 Twin (p)
Related Pages
•
•
•
•
•
•
•
Twin Primes from the World of Mathematics
Brun's Constant
Hardy-Littlewood Constants
Two Hardy-Littlewood Conjectures
The Prime Glossary's: Twin primes
A table of the number of twin primes to 1014
The chronology of prime number records (includes the record twins by year)
References
References:
Forbes97
T. Forbes, "A large pair of twin primes," Math. Comp., 66 (1997) 451455. MR 97c:11111
Abstract: We describe an efficient integer squaring
algorithm (involving the fast Fourier transform modulo
F8) that was used on a 486 computer to discover a large
pair of twin primes.
[The twin primes 6797727 · 215328± 1 are found on a 486
microcomputer]
IJ96
K. Indlekofer and A. Járai, "Largest known twin primes," Math.
Comp., 65 (1996) 427-428. MR 96d:11009
Abstract: The numbers 697053813 · 216352± 1 are twin
primes.
PSZ90
B. K. Parady, J. F. Smith and S. E. Zarantonello, "Largest known
twin primes," Math. Comp., 55 (1990) 381-382. MR 90j:11013
(Annotation available)