Twin prime

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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.)

The question of whether there exist infinitely many twin primes has been one of the great open questions in number theory for many years. 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.

Using his celebrated sieve method, Viggo Brun shows that the number of twin primes less than x is << x/(log x)². This result implies that the sum of the reciprocals of all twin primes converges (see Brun's constant and Brun's theorem). This is in stark contrast to the sum of the reciprocals of all primes, which diverges. He also shows that every even number can be represented in infinitely many ways as a difference of two numbers both having at most 9 prime factors. Chen Jingrun's well known theorem states that for any m even, there are infinitely many primes that differ by m from a number having at most two prime factors. (Before Brun attacked the twin prime problem, Jean Merlin (1876-1914) had also attempted to solve this problem using the sieve method. He was killed in World War I.)

Every twin prime pair greater than 3 is of the form (6n − 1, 6n + 1) for some natural number n, and with the exception of n = 1, n must end in 0, 2, 3, 5, 7, or 8.

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

$4((m-1)! + 1) = -m \mod (m(m+2))$

As of 2006, the largest known twin prime is 100314512544015 · 2¹⁷¹⁹⁶⁰ ± 1; it was found in 2006 by the Hungarians Zoltán Járai, Gabor Farkas, Timea Csajbok, Janos Kasza and Antal Járai. It has 51780 digits [1].

An empirical analysis of all prime pairs up to 4.35 · 10¹⁵ shows that the number of such pairs less than x is x·f(x)/(log x)² where f(x) is about 1.7 for small x and decreases to about 1.3 as x tends to infinity. The limiting value of f(x) is conjectured to equal the twin prime constant

$2 \prod_{p \geq 3} (1 - \frac{1}{(p-1)^2}) = 1.3203236\ldots;$

this conjecture would imply the twin prime conjecture, but remains unresolved.

[edit] 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)

Every third odd number is divisible by 3, so 5 is the only prime which is part of two pairs. The lower member of a pair is by definition a Chen prime.