Twin prime

Two prime numbers (p,q) are said to be twin primes if q=p+2. Except for the pair (2,3), this is the smallest possible distance between two primes. The first twin primes are:

   (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). 

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

<math>4((m-1)! + 1) = -m \ mod \ (m(m+2))</math>

(see factorial and modular arithmetic). Currently (2002), the largest known twin prime is 318032361 · 2¹⁰⁷⁰⁰¹±1; it was found in 2001 by Underbakke and Carmody using the free PrimeForm[?] software.

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