Twin prime conjecture
From Academic Kids

The twin prime conjecture is a famous problem in number theory that involves prime numbers. It states:
There are an infinite number of primes p such that p + 2 is also prime.
Such a pair of prime numbers is called a twin prime. The conjecture has been researched by many number theorists. Mathematicians believe the conjecture to be true, based only on numerical evidence and heuristic reasoning involving the probabilistic distribution of primes.
In 1849 de Polignac made the more general conjecture that for every natural number k, there are infinitely many prime pairs which have a distance of 2k. The case k = 1 is the twin prime conjecture.
Contents 
Partial results
In 1915, Viggo Brun showed that the sum of reciprocals of the twin primes was convergent. This famous result 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 <math>C N / \log^2 N<math> for some absolute constant C > 0.
In 1940, 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 Maier showed that a constant c < 0.25 can be used. In 2004 Goldston and Yıldırım showed that c could be improved further to 0.085786... In 2005, Goldston, Pintz and Yıldırım established that c can be chosen arbitrarily small[1] (http://www.arxiv.org/abs/math.NT/0505300), [2] (http://www.arxiv.org/abs/math.NT/0506067); in fact, if one assumes the ElliottHalberstam conjecture, they show that there are infinitely many n such that at least two of n, n+2, n+6, n+8, n+12, n+18, n+20 are prime.
In 1966, Chen Jingrun showed that there are infinitely many primes p such that p + 2 is either a prime or a semiprime (i.e., the product of two primes). The approach he took involved a topic called sieve theory, and he managed to treat the twin prime conjecture and Goldbach's conjecture in similar manners.
HardyLittlewood conjecture
There is also a generalization of the twin prime conjecture, known as the HardyLittlewood conjecture (after G. H. Hardy and John Littlewood), which is concerned with the distribution of twin primes, in analogy to the prime number theorem. Let π_{2}(x) denote the number of primes p ≤ x such that p + 2 is also prime. Define the twin prime constant C_{2} as
 <math>C_2 = \prod_{p\ge 3} \frac{p(p2)}{(p1)^2} \approx 0.66016118158468695739278121100145 ...<math>
(here the product extends over all prime numbers p ≥ 3). Then the conjecture is that
<math>\pi_2(x) \sim 2 C_2 \int_2^x {dt \over (\ln t)^2}<math>
in the sense that the quotient of the two expressions tends to 1 as x 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. The numerical evidence behind the HardyLittlewood conjecture is quite impressive.
Serious problem found in potential proof
On May 26, 2004, Richard Arenstorf of Vanderbilt University submitted a 38page proof that there are, in fact, infinitely many twin primes. On June 3, Michel Balazard of University Bordeaux reported that Lemma 8 on page 35 is false.[3] (http://listserv.nodak.edu/scripts/wa.exe?A2=ind0406&L=nmbrthry&F=&S=&P=1119) As is typical in mathematical proofs, the defect may be correctable or a substitute method may repair or replace the defect. Arenstorf withdrew his proof on June 8, noting "A serious error has been found in the paper, specifically, Lemma 8 is incorrect".
See also
External links
 "There Are Infinitely Many Prime Twins" by R. Arenstorf (http://arxiv.org/abs/math.NT/0405509) Withdrawn June 8, 2004.es:Conjetura de los n鷐eros primos gemelos
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