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, Ano

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1 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 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, Ano Number Digits Year Machine Prover 180(M 127 ) EDSAC1 Miller & Wheeler M SWAC Robinson (Jan 30) M SWAC Robinson (Jan 30) M SWAC Robinson (June 25) M SWAC Robinson (Oct 7) M SWAC Robinson (Oct 9) M BESK Riesel M , IBM7090 Hurwitz M , ILLIAC 2 Gillies M , ILLIAC 2 Gillies M , ILLIAC 2 Gillies M , IBM360/91 Tuckerman

2 M , M , CDC Cyber 174 CDC Cyber 174 Noll & Nickel Noll M , Cray 1 Nelson & Slowinski M , Cray 1 Slowinski M , Cray X-MP Slowinski M , Cray X-MP/24 Slowinski * , Amdahl 1200 Amdahl Six M , Cray-2 Slowinski & Gage M , Cray C90 Slowinski & Gage M , Cray T94 Slowinski & Gage M , M , M , M ,098, M ,053, M ,320, M ,235, M ,816, 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

3 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 ), 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 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 (and discovering the infamous pentium bug along the way), Thomas Nicely heuristically estimates Brun's constant to be 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!

4 Record Primes of this Type rank prime digits who when comment g Twin (p) g Twin (p) g Twin (p) p Twin (p) g Twin (p) g Twin (p) g Twin (p) p Twin (p) g Twin (p) g Twin (p) g Twin (p) g Twin (p) IJW 2000 Twin (p) IJW 2000 Twin (p) p Twin (p) IJW 1999 Twin (p) IJW 1999 Twin (p) g Twin (p) g Twin (p) g 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 The chronology of prime number records (includes the record twins by year) References References: Forbes97

5 T. Forbes, "A large pair of twin primes," Math. Comp., 66 (1997) 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 ± 1 are found on a 486 microcomputer] IJ96 K. Indlekofer and A. Járai, "Largest known twin primes," Math. Comp., 65 (1996) MR 96d:11009 Abstract: The numbers ± 1 are twin primes. PSZ90 B. K. Parady, J. F. Smith and S. E. Zarantonello, "Largest known twin primes," Math. Comp., 55 (1990) MR 90j:11013 (Annotation available)