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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, which states There are infinitely many primes p such that p + 2 is also prime. In 1849 de Polignac made the more general conjecture that for every natural number k, there are infinitely many prime pairs p and p′ such that p′ − p = 2k. The case k = 1 is the twin prime conjecture.

A stronger form of the twin prime conjecture, the Hardy–Littlewood conjecture, postulates a distribution law for twin primes akin to the prime number theorem.

In 1915, Viggo Brun showed that the sum of reciprocals of the twin primes was convergent. This famous result, called Brun's theorem, was the first use of the Brun sieve and helped initiate the development of modern sieve theory. The modern version of Brun's argument can be used to show that the number of twin primes less than N does not exceed

\frac{CN}{\log^2{N}}

for some absolute constant C > 0.

In 1940, Paul Erdős showed that there is a constant c < 1 and infinitely many primes p such that (p′ − p) < (c ln p) where p′ denotes the next prime after p. This result was successively improved; in 1986 Helmut Maier showed that a constant c < 0.25 can be used. In 2004 Daniel Goldston and Cem Yıldırım showed that the constant could be improved further to c = 0.085786… In 2005, Goldston, János Pintz and Yıldırım established that c can be chosen to be arbitrarily small[1][2]

\liminf_{n\to\infty}\frac{p_{n+1}-p_n}{\log p_n}=0.

In fact, by assuming the Elliott–Halberstam conjecture or a slightly weaker version, they were able to show that there are infinitely many n such that at least two of n, n + 2, n + 6, n + 8, n + 12, n + 18, or n + 20 are prime. Under a stronger hypothesis they showed that at least two of n, n + 2, n + 4, and n + 6 are prime.

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.

Contents

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 France.

On August 6, 2009 those same two projects announced that a new record twin prime had been found.[3] It is 65516468355 · 2333333 ± 1.[4] The numbers have 100355 decimal digits.

An empirical analysis of all prime pairs up to 4.35 · 1015 shows that if the number of such pairs less than x is f(xx/(log x)2 then f(x) is about 1.7 for small x and decreases towards about 1.3 as x tends to infinity.

There are 808,675,888,577,436 twin prime pairs below 1018.[5]

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 twin prime conjecture would give a better approximation, as with the prime counting function, by

\pi_2(x) \approx 2C_2\; \operatorname{li}_2(x) = 2C_2 \int_2^x \frac{dt}{\left(\log_e t \right)^2}.

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.

First Hardy–Littlewood conjecture

The Hardy–Littlewood conjecture (after G. H. Hardy and John Littlewood) is a generalization of the twin prime conjecture. It is concerned with the distribution of prime constellations, including twin primes, in analogy to the prime number theorem. Let π2(x) denote the number of primes px such that p + 2 is also prime. Define the twin prime constant C2 as[6]

C_2 = \prod_{p\ge 3} \frac{p(p-2)}{(p-1)^2} \approx 0.66016 18158 46869 57392 78121 10014\dots

(here the product extends over all prime numbers p ≥ 3). Then the conjecture is that

\pi_2(n) \sim 2 C_2 \frac{n}{(\ln n)^2} \sim 2 C_2 \int_2^n {dt \over (\ln t)^2}

in the sense that the quotient of the two expressions tends to 1 as n approaches infinity.

This conjecture can be justified (but not proven) by assuming that 1 / ln t describes the density function of the prime distribution, an assumption suggested by the prime number theorem.

Polignac's conjecture

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

External links

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Simple English

The twin prime conjecture is a mathematical theory. It says that it is possible to find two twin primes that are as big as wanted.

Twin primes are prime numbers that differ by two. For example 3 and 5 are both prime and differ by two. They are twin primes. 23 is prime, but it is not a twin prime. The primes nearest to 23 are 19 and 29. Twin primes were discovered by Euclid in 300 B.C.

Since Euclid's time mathematicians have wondered whether there are an infinite number of twin primes. Many mathematicians are still trying to find the answer.


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