## twin primesPairs 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 {6n-1, 6n+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. ## Related category• PRIME NUMBERS | |||||

Home • About • Copyright © The Worlds of David Darling • Encyclopedia of Alternative Energy • Contact |