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) |
(2) |
Define
(3) |
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) |
(5) |
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.
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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.
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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.
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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.
© 1996-9 Eric W. Weisstein