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 , there are infinitely many prime pairs which have a distance of . The case is the twin prime conjecture.

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

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