Twin prime

A twin prime (or prime twin) is a pair of prime numbers whose difference is two. Except for the pair (2,3), this is the smallest possible difference between two primes. Some examples of twin primes are 5 and 7, 11 and 13, and 821 and 823.

It is unknown whether there exist infinitely many twin primes, but most number theorists believe this to be true. This is the content of the Twin Prime Conjecture. A strong form of the Twin Prime Conjecture, the Hardy-Littlewood conjecture, postulates a distribution law for twin primes akin to the prime number theorem.

It is known that the sum of the reciprocals of all twin primes converges (see Brun's constant). This is in stark contrast to the sum of the reciprocals of all primes, which diverges.

Every twin prime pair greater than 3 is of the form 6n - 1, 6n +1 for some natural number n.

One can prove that the pair m, m + 2 is a twin prime if and only if

Currently (2003), the largest known twin prime is 33218925 · 2¹⁶⁹⁶⁹⁰±1; it was found in 2002 by Papp using the free Proth and NewPGen software.

Table of contents

1 The first 35 twin primes
2 Related Articles
3 External links

The first 35 twin primes

  (3,  5),    (5,  7),    (11, 13),   (17, 19),   (29, 31),   (41, 43),   (59, 61), 
  (71,  73),  (101, 103), (107, 109), (137, 139), (149, 151), (179, 181), (191, 193),
  (197, 199), (227, 229), (239, 241), (269, 271), (281, 283), (311, 313), (347, 349),
  (419, 421), (431, 433), (461, 463), (521, 523), (569, 571), (599, 601), (617, 619),
  (641, 643), (659, 661), (809, 811), (821, 823), (827, 829), (857, 859), (881, 883).

External links

Chris Caldwell: Twin Primes, http://www.utm.edu/research/primes/lists/top20/twin.html
Xavier Gourdon, Pascal Sebah: Introduction to Twin Primes and Brun's Constant, http://numbers.computation.free.fr/Constants/Primes/twin.html