twin prime conjecture

Two consecutive odd numbers which are both prime are called twin primes, 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 infinite number of twin primes ?

In 1849 de Polignac made the more general conjecture that for every natural number $n$, 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 and infinitely many primes $p$ such that where $q$ denotes the next prime after $p$. This result was improved in 1986 by Maier; he showed that a constant 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