# twin primes

Twin primes are pairs of prime numbers that differ by
two, the first of which are 3 and 5, 5 and 7, 11 and 13, and 17 and 19.
The largest example known, as of February 2003, is a pair of 51,090-digit
primes discovered by Yves Gallot and Daniel Papp, with the value 33218925
× 2^{169690}±1. Other than the first, all twin primes have
the form {6*n*-1, 6*n*+1}; also, the integers *n* and *n* + 2 form twin primes if and only if 4[(*n* -1 )! + 1]
= -*n* (mod *n*(*n* + 2)).

In 1919 Brun showed that the sum of the reciprocals of the twin primes converges
to a sum now known as *Brun's constant*: (1/3 + 1/5) + (1/5 + 1/7)
+ (1/11 + 1/13) + (1/17 + 1/19) + ... In 1994, by calculating the twin primes
up to 10^{14}, (and discovering the infamous Pentium bug in the
process) Thomas Nicely of Lynchburg College estimated Brun's constant to
be 1.902160578.

According to the (unsolved) **twin-prime conjecture** there
are infinitely many twin primes. The twin-prime conjecture generalizes to
prime pairs that differ by any even number *n*, and generalizes even
further to certain finite patterns of numbers separated by specified even
differences. For example, the following triplets of primes all fit the pattern *k*, *k* + 2, and *k* + 6: 5, 7, and 11; 11, 13, and
17; 17, 19, and 23; 41, 43, and 47. It is believed that for any such pattern
not outlawed by divisibility considerations there are infinitely many examples.
(The pattern *k*, *k* + 2, and *k* + 4 has only one
solution in primes, 3, 5, and 7, because any larger such triplet would contain
a number divisible by 3.) Quartets of the form *k*, *k* +
2, *k* + 6, and *k* + 8 (the smallest example is 5, 7, 11,
and 13) are thought to be infinite. For some patterns no example is known,
or only one.