“Twin Primes”的概念、定义、习题解题方法及翻译-数学辞典

单词

Twin Primes

释义

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 A001359and A006512).

Let be the number of twin primes and such that . It is not known if there are an infinitenumber of such Primes (Shanks 1993), but all twin primes except (3, 5) are of the form . J. R. Chen has shownthere 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 reducedto 6.8325 (Haugland 1999). This calculation involved evaluation of 7-fold integrals andfitting 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 Davenport1966), and

(Pil'Tai 1972), as quoted in Le Lionnais (1983, p. 26).

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

()	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 IntelPentium microprocessor by computing the reciprocals of 824,633,702,441 and 824,633,702,443, which should have beenaccurate 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 upto

in his calculation of Brun's Constant. The following table gives the number less than increasing powersof 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, TwinPrime Conjecture, Twin Primes Constant

References

Bombieri, E. and Davenport, H. ``Small Differences Between Prime Numbers.'' Proc. Roy. Soc. Ser. A 293, 1-8, 1966.

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. SequencesA014574,A001359/M2476,A006512/M3763, andA007508/M1855in ``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.

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