#### Twin Primes

Twin primes are a pair of primes of the form (, ). 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).

The following table gives the first few for the twin primes (, ), cousin primes (, ), sexy primes (, ), etc.

 Triplet Sloane First Member (, ) Sloane's A001359 3, 5, 11, 17, 29, 41, 59, 71, ... (, ) Sloane's A023200 3, 7, 13, 19, 37, 43, 67, 79, ... (, ) Sloane's A023201 5, 7, 11, 13, 17, 23, 31, 37, ... (, ) Sloane's A023202 3, 5, 11, 23, 29, 53, 59, 71, ... (, ) Sloane's A023203 3, 7, 13, 19, 31, 37, 43, 61, ... (, ) Sloane's A046133 5, 7, 11, 17, 19, 29, 31, 41, ...

Let be the number of twin primes and such that . It is not known if there are an infinite number of such primes (Wells 1986, p. 41; Shanks 1993), but it seems almost certain to be true (Hardy and Wright 1979, p. 5). 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 1996, p. 261). It has been shown that
 (2)

written more concisely as
 (3)

where is known as the twin primes constant and is another constant. The constant 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 1996, p. 262).

Wolf notes that the formula

 (4)

which increases as for large , agrees with numerical data much better than does , although not as well as .

Extending the search done by Brent in 1974 or 1975, Wolf has searched for the analog of the Skewes number for twins, i.e., an such that changes sign. Wolf checked numbers up to and found more than 90,000 sign changes. From this data, Wolf conjectured that the number of sign changes for of is given by

 (5)

Proof of this conjecture would also imply the existence an infinite number of twin primes.

Define

 (6)

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.

 () 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

 (7)

where is a pair of twin primes iff
 (8)

(Ribenboim 1996, p. 259). S. M. Ruiz has found the unexpected result that are twin primes iff

 (9)

for , where is the floor function.

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; Nicely 1998, 1999). Using a distributed computation, Fry et al. obtained in 2000, although this value has not yet been made public. The following table gives for various values of , and extends a similar table with early references given by Ribenboim (1996, p. 263).

 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 1,177,209,242,304

It is conjectured that every even number is a sum of a pair of twin primes except a finite number of exceptions whose first few terms are 2, 4, 94, 96, 98, 400, 402, 404, 514, 516, 518, ... (Sloane's A007534; Wells 1986, p. 132).