## Twin Primes

Twin primes are Primes (, ) such that . The first few twin primes are for , 6, 12, 18, 30, 42, 60, 72, 102, 108, 138, 150, 180, 192, 198, 228, 240, 270, 282, ... (Sloane's A014574). Explicitly, these are (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), ... (Sloane's A001359 and A006512).

Let be the number of twin primes and such that . It is not known if there are an infinite number of such Primes (Shanks 1993), but all twin primes except (3, 5) are of the form . J. R. Chen has shown there exists an Infinite number of Primes such that has at most two factors (Le Lionnais 1983, p. 49). Bruns proved that there exists a computable Integer such that if , then

 (1)

(Ribenboim 1989, p. 201). It has been shown that
 (2)

where has been reduced to (Fouvry and Iwaniec 1983), (Fouvry 1984), 7 (Bombieri et al. 1986), 6.9075 (Fouvry and Grupp 1986), and 6.8354 (Wu 1990). The bound on is further reduced to 6.8325 (Haugland 1999). This calculation involved evaluation of 7-fold integrals and fitting of three different parameters. Hardy and Littlewood conjectured that (Ribenboim 1989, p. 202).

Define

 (3)

If there are an infinite number of twin primes, then . The best upper limit to date is (Huxley 1973, 1977). The best previous values were 15/16 (Ricci), (Bombieri and Davenport 1966), and (Pil'Tai 1972), as quoted in Le Lionnais (1983, p. 26).

Some large twin primes are , , and . An up-to-date table of known twin primes with 2000 or more digits follows. An extensive list is maintained by Caldwell.

 () Digits Reference 2003 Atkin and Rickert 1984 2009 Dubner, Atkin 1985 2151 Dubner 1992 2259 Dubner, Atkin 1985 2309 Brown et al. 1989 2309 Dubner 1989 2324 Brown et al. 1989 2500 Dubner 1991 2571 Dubner 1993 3389 Noll et al. 1989 3439 Dubner 1993 4030 Dubner 1993 4622 Forbes 1995 4932 Indlekofer and Ja'rai 1994 5129 Dubner 1995 11713 Indlekofer and Ja'rai 1995

The last of these is the largest known twin prime pair. In 1995, Nicely discovered a flaw in the Intel Pentium microprocessor by computing the reciprocals of 824,633,702,441 and 824,633,702,443, which should have been accurate to 19 decimal places but were incorrect from the tenth decimal place on (Cipra 1995, 1996; Nicely 1996).

If , the Integers and form a pair of twin primes Iff

 (4)

where is a pair of twin primes Iff
 (5)

(Ribenboim 1989). The values of were found by Brent (1976) up to . T. Nicely calculated them up to in his calculation of Brun's Constant. The following table gives the number less than increasing powers of 10 (Sloane's A007508).

 35 205 1224 8,169 58,980 440,312 3,424,506 27,412,679 224,376,048 1,870,585,220 15,834,664,872 135,780,321,665

See also Brun's Constant, de Polignac's Conjecture Prime Constellation, Sexy Primes, Twin Prime Conjecture, Twin Primes Constant

References

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Bombieri, E.; Friedlander, J. B.; and Iwaniec, H. Primes in Arithmetic Progression to Large Moduli.'' Acta Math. 156, 203-251, 1986.

Bradley, C. J. The Location of Twin Primes.'' Math. Gaz. 67, 292-294, 1983.

Brent, R. P. Irregularities in the Distribution of Primes and Twin Primes.'' Math. Comput. 29, 43-56, 1975.

Brent, R. P. UMT 4.'' Math. Comput. 29, 221, 1975.

Brent, R. P. Tables Concerning Irregularities in the Distribution of Primes and Twin Primes to .'' Math. Comput. 30, 379, 1976.

Caldwell, C. http://www.utm.edu/cgi-bin/caldwell/primes.cgi/twin.

Cipra, B. How Number Theory Got the Best of the Pentium Chip.'' Science 267, 175, 1995.

Cipra, B. Divide and Conquer.'' What's Happening in the Mathematical Sciences, 1995-1996, Vol. 3. Providence, RI: Amer. Math. Soc., pp. 38-47, 1996.

Fouvry, É. Autour du théorème de Bombieri-Vinogradov.'' Acta. Math. 152, 219-244, 1984.

Fouvry, É. and Grupp, F. On the Switching Principle in Sieve Theory.'' J. Reine Angew. Math. 370, 101-126, 1986.

Fouvey, É. and Iwaniec, H. Primes in Arithmetic Progression.'' Acta Arith. 42, 197-218, 1983.

Guy, R. K. Gaps between Primes. Twin Primes.'' §A8 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 19-23, 1994.

Haugland, J. K. Application of Sieve Methods to Prime Numbers. Ph.D. thesis. Oxford, England: Oxford University, 1999.

Huxley, M. N. Small Differences between Consecutive Primes.'' Mathematica 20, 229-232, 1973.

Huxley, M. N. Small Differences between Consecutive Primes. II.'' Mathematica 24, 142-152, 1977.

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 1983.

Nicely, T. R. The Pentium Bug.'' http://lasi.lynchburg.edu/Nicely_T/public/pentbug/pentbug.htm.

Nicely, T. Enumeration to of the Twin Primes and Brun's Constant.'' Virginia J. Sci. 46, 195-204, 1996. http://lasi.lynchburg.edu/Nicely_T/public/twins/twins.htm.

Parady, B. K.; Smith, J. F.; and Zarantonello, S. E. Largest Known Twin Primes.'' Math. Comput. 55, 381-382, 1990.

Ribenboim, P. The Book of Prime Number Records, 2nd ed. New York: Springer-Verlag, pp. 199-204, 1989.

Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 30, 1993.

Sloane, N. J. A. Sequences A014574, A001359/M2476, A006512/M3763, and A007508/M1855 in An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

Weintraub, S. A Prime Gap of 864.'' J. Recr. Math. 25, 42-43, 1993.

Wu, J. Sur la suite des nombres premiers jumeaux.'' Acta. Arith. 55, 365-394, 1990.