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Encyclopedia > Twin prime conjecture

The twin prime conjecture is a famous problem in number theory that involves prime numbers. It was first proposed by Euclid around 300 B.C. and states: Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ... Euclid (Greek: ), also known as Euclid of Alexandria, was a Greek mathematician who flourished in Alexandria, Egypt, almost certainly during the reign of Ptolemy I (323–283 BC). ...

There are infinitely many primes p such that p + 2 is also prime.

Such a pair of prime numbers 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 and heuristic reasoning involving the probabilistic distribution of primes using Cramér's model. In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ... A twin prime is a prime number that differs from another prime number by two. ... Look up Heuristic in Wiktionary, the free dictionary. ... In mathematics, Cramérs conjecture, formulated by the Swedish mathematician Harald Cramér in 1937, states that where pn denotes the nth prime number and log is the natural logarithm. ...


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. Alphonse de Polignac (1817 – 1890) was a French mathematician. ...

Contents

Partial results

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 Viggo Brun (October 13, 1882 - August 15, 1978) was a Norwegian mathematician. ... In mathematics, Bruns theorem is a result of Viggo Brun in number theory. ... Sieve theory is a set of general techniques in number theory, designed to count, or more realistically to estimate the size of, sifted sets of integers. ...

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 arbitrarily small [1], [2]: Paul ErdÅ‘s, also Pál ErdÅ‘s, in English Paul Erdos or Paul Erdös (March 26, 1913 – September 20, 1996), was an immensely prolific (and famously eccentric) Hungarian mathematician who, with hundreds of collaborators, worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set... Helmut Maier is an American mathematician who made significant progress in the study of Twin prime conjecture. ... Daniel Alan Goldston is an American mathematician who specializes in number theory. ... Cem Yalcin Yildirim is a Turkish mathematician who specializes in number theory. ...

liminf_{ntoinfty}frac{p_{n+1}-p_n}{log p_n}=0

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, n + 20 are prime. In number theory, the Elliott-Halberstam conjecture is a conjecture about the distribution of prime numbers in arithmetic progressions. ...


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 sieve theory, and he managed to treat the twin prime conjecture and Goldbach's conjecture in similar manners. Chen Jingrun (ch. ... In mathematics, a semiprime (also called biprime or 2-almost prime, or pq number) is a natural number that is the product of two (not necessarily distinct) prime numbers. ... Goldbachs conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. ...


Defining a Chen prime to be a prime p such that p + 2 is either a prime or a semiprime, Terence Tao and Ben Green showed in 2005 that there are infinitely many three-term arithmetic progressions of Chen primes. A prime number p is called a Chen prime if p + 2 is either a prime or a product of two primes. ... In mathematics, a semiprime (also called biprime or 2-almost prime, or pq number) is a natural number that is the product of two (not necessarily distinct) prime numbers. ... Terence Chi-Shen Tao ( 陶哲軒)(born 1975) is an Australian mathematician working primarily on harmonic analysis, partial differential equations, combinatorics, analytic number theory and representation theory. ... Ben Joseph Green (born February 27, 1977, Bristol, United Kingdom) is a British mathematician, specializing in combinatorics and number theory. ...


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 G. H. Hardy Professor Godfrey Harold Hardy FRS (February 7, 1877 – December 1, 1947) was a prominent English mathematician, known for his achievements in number theory and mathematical analysis. ... John Edensor Littlewood (June 9, 1885 – September 6, 1977) was a British mathematician. ... In number theory, the prime number theorem (PNT) describes the approximate, asymptotic distribution of the prime numbers. ...

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

(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 x approaches infinity. In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as their index increases indefinitely. ...


This conjecture can be justified (but not proven) by assuming that

frac{1}{ln{t}}

describes the density function of the prime distribution, an assumption suggested by the prime number theorem. The numerical evidence behind the Hardy–Littlewood conjecture is quite impressive. In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ...


See also

A twin prime is a prime number that differs from another prime number by two. ... In mathematics, Bruns theorem is a result of Viggo Brun in number theory. ... Goldbachs conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. ... In mathematics, Cramérs conjecture, formulated by the Swedish mathematician Harald Cramér in 1937, states that where pn denotes the nth prime number and log is the natural logarithm. ...

External links

  • NOVA Science Now on the Twin Prime Conjecture
  • Terrence Tao on the difficulty of proving the twin primes conjecture]

  Results from FactBites:
 
Conjecture (448 words)
In mathematics, a conjecture is a mathematical statement which has been proposed as a true statement, but which no one has been able to prove or disprove.
When a conjecture has been proven to be true, it becomes known as a theorem, and joins the realm of mathematical facts.
Although many of the most famous conjectures have been tested across an astounding range of numbers, this is no guarantee against a single counterexample, which would immediately disprove the conjecture.
Twin prime (196 words)
It is unknown whether there exist infinitely many twin primes, but most number theorists believe this to be true.
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.
It is known that the sum of the reciprocals of all twin primes converges (see Brun's constant).
  More results at FactBites »

 
 

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