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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 (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), and (skipping quite a few), (821, 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 methods, Viggo Brun shows that the number of twin primes less than x is O(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 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.)

Every twin prime pair except (3, 5) 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 proved that the pair m, m + 2 is a twin prime if and only if

4((m-1)! + 1) \equiv -m \pmod {m(m+2)}.


If m − 4 or m + 6 is also prime then the 3 primes are called a prime triplet.

Largest known twin prime

On January 15, 2007 two distributed computing projects, Twin Prime Search and PrimeGrid found the largest known twin primes, 2003663613 · 2195000 ± 1. The numbers have 58711 decimal digits. Their discoverer was Eric Vautier of Francemarker.

On August 6, 2009 those same two projects announced that a new record twin prime had been found. It is 65516468355 · 2333333 ± 1. The numbers have 100355 decimal digits.

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 twice the twin prime constant (not to be confused with Brun's constant)
2 \prod_{\textstyle{p\;{\rm prime}\atop p \ge 3}} \left(1 - \frac{1}{(p-1)^2}\right) = 1.3203236\ldots;
this conjecture would imply the twin prime conjecture, but remains unresolved.

The first 35 twin prime pairs

There are 35 twin prime pairs below 1000, given in the following list:
(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 greater than seven 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.

Polignac's conjecture from 1849 states that for every even natural number k, there are infinitely many prime pairs p and p′ such that p − p′ = k. The case k = 2 is the twin prime conjecture. The case k = 4 corresponds to cousin primes and the case k = 6 to sexy primes. The conjecture has not been proved or disproved for any value of k.

See also



References



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