Chen's theorem on primes and semiprimes with fixed separation

Statement

For every positive even integer $h$, there exist infinitely many pairs $(p,p+h)$ such that $p$ is a prime number and $p+h$ is either a prime or a semiprime.

The theorem applies in particular to $h=2$, showing that there are infinitely many pairs of the form $(p,p+2)$ where $p$ is a prime and $p+2$ is either prime or semiprime.

Relation with other facts/conjectures

• Twin primes conjecture: This states that there are infinitely many pairs with a prime gap of $2$.