Twin primes
The term was coined by Paul Stackel (1862-1919).
The first few twin prime pairs are:
(3,5), (5,7), (11,13), (17,19), (29,31), (41,43), /59,61), (71,73), (101,103), (107,109), (137,139) ...
The smallest gap possible between primes is 2, the only exception is (2,3).
The twin prime (3,5) is the only one in the form n + 2; all the others are of the form 6n + 1.
An important result for twin primes is Brun's theorem, which states that the number obtained by adding the reciprocals of the odd twin primes converges to a define number
Twin primes appear despite the general tendency of gaps between adjacent primes to become larger as the numbers themselves get larger.
The question of whether there exist infinitely twin primes is an open question in the number theory.
The twin prime conjecture states: There are infinitely many primes p such that p + 2 is also prime.
On December 25, 2011 the largest known twin primes had been found. I
t is 3756801695685 * 2^666669 + 1.
The number have 200700 decimal digits.