Green-Tao theorem

History

This result is a special case of the Erdős conjecture on arithmetic progressions, which states that any large set contains aribtrarily long arithmetic progression. (It is a special case because the set of primes is large).

Statement

This states that for any positive integer $k$, there exists a prime arithmetic progression of length $k$, i.e., an arithmetic progression of length $k$ all of whose members are prime numbers.

Relation with other facts/conjectures

Stronger facts/conjectures

• Erdős conjecture on arithmetic progressions
• Tao-Ziegler theorem on primes in polynomial progressions
• Zhou's theorem on arbitrarily large arithmetic progressions of Chen primes