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 M n = 2 n 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 2000 Ano Number Digits Year Machine Prover 180(M 127 ) 2 +1 79 1951 EDSAC1 Miller & Wheeler M 521 157 1952 SWAC Robinson (Jan 30) M 607 183 1952 SWAC Robinson (Jan 30) M 1279 386 1952 SWAC Robinson (June 25) M 2203 664 1952 SWAC Robinson (Oct 7) M 2281 687 1952 SWAC Robinson (Oct 9) M 3217 969 1957 BESK Riesel M 4423 1,332 1961 IBM7090 Hurwitz M 9689 2,917 1963 ILLIAC 2 Gillies M 9941 2,993 1963 ILLIAC 2 Gillies M 11213 3,376 1963 ILLIAC 2 Gillies M 19937 6,002 1971 IBM360/91 Tuckerman
M 21701 6,533 1978 M 23209 6,987 1979 CDC Cyber 174 CDC Cyber 174 Noll & Nickel Noll M 44497 13,395 1979 Cray 1 Nelson & Slowinski M 86243 25,962 1982 Cray 1 Slowinski M 132049 39,751 1983 Cray X-MP Slowinski M 216091 65,050 1985 Cray X-MP/24 Slowinski 391581*2 216193-1 65,087 1989 Amdahl 1200 Amdahl Six M 756839 227,832 1992 Cray-2 Slowinski & Gage M 859433 258,716 1994 Cray C90 Slowinski & Gage M 1257787 378,632 1996 Cray T94 Slowinski & Gage M 1398269 420,921 1996 M 2976221 895,932 1997 M 3021377 909,526 1998 M 6972593 2,098,960 1999 M 13466917 4,053,946 2001 M 20996011 6,320,430 2003 M 24036583 7,235,733 2004 M 25964951 7,816,230 2005 Pentium (90 Pentium (100 Pentium (200 Pentium (350 AMD T-Bird (800 Pentium (2 GHz) Pentium 4 (2.4GHz) Pentium 4 (2.4GHz) Armengaud, Woltman, et. al. [GIMPS] Spence, Woltman, et. al. [GIMPS] Clarkson, Woltman, Hajratwala, Woltman, Cameron, Woltman, Shafer, Woltman, Kurowski, et. al. [GIMPS, Findley, GIMPS et. al. 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 10 6 8169 8248 10 8 440312 440368 10 10 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 10 14 (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 prime digits who when comment 1 33218925 2 169690-1 51090 g259 2002 Twin (p) 2 60194061 2 114689-1 34533 g294 2002 Twin (p) 3 1765199373 2 107520-1 32376 g182 2002 Twin (p) 4 318032361 2 107001-1 32220 p100 2001 Twin (p) 5 1807318575 2 98305-1 29603 g216 2001 Twin (p) 6 665551035 2 80025-1 24099 g216 2000 Twin (p) 7 1940734185 2 66445-1 20012 g336 2004 Twin (p) 8 781134345 2 66445-1 20011 p53 2001 Twin (p) 9 1693965 2 66443-1 20008 g183 2000 Twin (p) 10 83475759 2 64955-1 19562 g144 2000 Twin (p) 11 37831341 2 61777-1 18605 g277 2003 Twin (p) 12 291889803 2 60090-1 18098 g191 2001 Twin (p) 13 4648619711505 2 60000-1 18075 IJW 2000 Twin (p) 14 2409110779845 2 60000-1 18075 IJW 2000 Twin (p) 15 488162409 2 53333-1 16064 p135 2004 Twin (p) 16 2230907354445 2 48000-1 14462 IJW 1999 Twin (p) 17 871892617365 2 48000-1 14462 IJW 1999 Twin (p) 18 160675905 2 42004-1 12653 g350 2003 Twin (p) 19 1186468455 2 41537-1 12513 g294 2003 Twin (p) 20 1902519837 2 41536-1 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 10 14 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) 451-455. MR 97c:11111 Abstract: We describe an efficient integer squaring algorithm (involving the fast Fourier transform modulo F 8 ) that was used on a 486 computer to discover a large pair of twin primes. [The twin primes 6797727 2 15328 ± 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 2 16352 ± 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)