twin prime conjecture

Two consecutive odd numbersMathworldPlanetmathPlanetmath which are both prime are called twin primesMathworldPlanetmath, e.g. 5 and 7, or 41 and 43, or 1,000,000,000,061 and 1,000,000,000,063. But is there an infiniteMathworldPlanetmath number of twin primes ?

In 1849 de Polignac made the more general conjecture that for every natural numberMathworldPlanetmath n, there are infinitely many prime pairs which have a distance of 2n. The case n=1 is the twin prime conjecture.

In 1940, Erdős showed that there is a constant c<1 and infinitely many primes p such that q-p<clnp where q denotes the next prime after p. This result was improved in 1986 by Maier; he showed that a constant c<0.25 can be used. The constant c is called the twin prime constant.

In 1966, Chen Jingrun showed that there are infinitely many primes p such that p+2 is either a prime or a semiprime.

Title twin prime conjecture
Canonical name TwinPrimeConjecture
Date of creation 2013-03-22 13:21:32
Last modified on 2013-03-22 13:21:32
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 11
Author alozano (2414)
Entry type Conjecture
Classification msc 11N05
Related topic PrimeTriplesConjecture
Related topic BrunsConstant
Defines twin prime constant
Defines twin primes