- Twin prime conjecture
The twin prime conjecture is a famous unsolved problem in
number theory that involvesprime number s. It states::"There are infinitely many primes" "p" "such that" "p" + 2 "is also prime."
Such a pair of
prime number s is called a prime twin. The conjecture has been researched by many number theorists. Mathematicians believe the conjecture to be true, based only on numerical evidence andheuristic reasoning involving the probabilistic distribution of primes using theCramér model .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" = 2"k". The case "k" = 1 is the twin prime conjecture.
Partial results
In 1915,
Viggo Brun showed that the sum of reciprocals of the twin primes was convergent. This famous result, calledBrun's theorem , was the first use of theBrun sieve and helped initiate the development of modernsieve 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:
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 1986Helmut Maier showed that a constant "c" < 0.25 can be used. In 2004Daniel Goldston andCem 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 [cite web
url = http://www.arxiv.org/abs/math.NT/0505300
title = Small Gaps between Primes Exist (article abstract)
accessdate = 2007-06-20
date = 2007] [cite web
url = http://www.arxiv.org/abs/math.NT/0506067
title = Small gaps between primes or almost primes (article abstact)
accessdate = 2007-06-20
date = 2007]:
In fact, by assuming the
Elliott–Halberstam conjecture , 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.In 1966,
Chen Jingrun showed that there are infinitely many primes "p" such that "p" + 2 is either a prime or asemiprime (i.e., the product of two primes). The approach he took involved sieve theory, and he managed to treat the twin prime conjecture andGoldbach's conjecture in similar manners (seeChen's theorem ).Defining a
Chen prime to be a prime "p" such that "p" + 2 is either a prime or asemiprime ,Terence Tao andBen J. Green showed in 2005 that there are infinitely many three-term arithmetic progressions of Chen primes.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 ofprime constellation s, including twin primes, in analogy to theprime number theorem . Let π2("x") denote the number of primes "p" ≤ "x" such that "p" + 2 is also prime. Define the twin prime constant "C"2 as [cite web
url = http://www.gn-50uma.de/alula/essays/Moree/Moree-details.en.shtml#t05-twin
title = A page of number theoretical constants
accessdate = 2007-06-20
date = 2007]:
(here the product extends over all prime numbers "p" ≥ 3). Then the conjecture is that
:
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
:
describes the
density function of the prime distribution, an assumption suggested by the prime number theorem.ee also
*
Arithmetic derivative
*Prime gap References
External links
* [http://www.pbs.org/wgbh/nova/sciencenow/3302/02.html NOVA Science Now on the Twin Prime Conjecture]
* [http://terrytao.wordpress.com/2007/04/05/simons-lecture-i-structure-and-randomness-in-fourier-analysis-and-number-theory/ Terrence Tao on the difficulty of proving the twin primes conjecture]
*MathWorld|urlname=TwinPrimeConjecture|title=Twin Prime Conjecture
*MathWorld|urlname=TwinPrimesConstant|title=Twin Primes Constant
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