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, Erdos showed that there is a constant $c<1$ and infinitely many primes $p$ such that $q-p<c \ln{p}$ 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.