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.
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
Wolf notes that the formula
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
Some large twin primes are , , and . An up-to-date table of known
twin primes with 2000 or more digits follows.
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
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).
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).