A **twin prime** is a prime number that differs from another prime number by two. Except for the pair (2, 3), this is the smallest possible difference between two primes. Some examples of twin prime pairs are 5 and 7, 11 and 13, and 821 and 823. (Sometimes the term *twin prime* is used for a pair of twin primes; an alternative name for this is **prime twin**.)

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 (6*n* - 1, 6*n* + 1) for some natural number *n*.

It has been proven that the pair *m*, *m* + 2 is a twin prime if and only if

^{169690}± 1; it was found in 2002 by Papp using the free Proth and NewPGen software.

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## The first 35 twin prime pairs

(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)