Green–Tao theorem

Remarks

Green and Tao proved more: every subset $A$ of the set $\mathcal{P}$ of primes with $$\limsup_{N \to \infty} \frac{\abs{A \cap \set{1, \dots, N}}}{\abs{\mathcal{P} \cap \set{1, \dots, N}}} > 0$$ contains arithmetic progressions of every length. Szemerédi's theorem does not apply directly, because the primes have density zero by the prime number theorem. The proof is a transference principle. A relative Szemerédi theorem holds for dense subsets of a pseudorandom set. The primes sit densely inside such a set, built from sieve weights of Goldston and Yıldırım. For example, $5, 11, 17, 23, 29$ is a progression of five primes with difference $6$. The primes contain no infinite arithmetic progression: for $a \ge 2$ the term $a + ad = a(1 + d)$ is composite, and for $a = 1$ the first term is not prime.

Going deeper

For a student

The Green–Tao theorem combines additive combinatorics with analytic number theory. Its idea is that Szemerédi's theorem survives inside sparse but random-looking sets. It is a special case of the open Erdős conjecture on arithmetic progressions, since the sum of the reciprocals of the primes diverges.