# Green–Tao theorem

In number theory, the Green–Tao theorem, proved by Ben Green and Terence Tao in 2004,[1] states that the sequence of prime numbers contains arbitrarily long arithmetic progressions. In other words there exist arithmetic progressions of primes, with k terms, where k can be any natural number. The proof is an extension of Szemerédi's theorem.

In 2006, Terence Tao and Tamar Ziegler extended the result to cover polynomial progressions.[2] More precisely, given any integer-valued polynomials P1,..., Pk in one unknown m all with constant term 0, there are infinitely many integers x, m such that x + P1(m), ..., x + Pk(m) are simultaneously prime. The special case when the polynomials are m, 2m, ..., km implies the previous result that there are length k arithmetic progressions of primes.

## Statement

Let $\pi(N)$ denote the number of primes less than or equal to $N$. If $A$ is a subset of the prime numbers such that

$\limsup_{N\rightarrow\infty} \dfrac{|A\cap [1,N]|}{\pi(N)}>0$,

then for all positive integers $k$, the set $A$ contains infinitely many arithmetic progressions of length $k$.

This statement implies that the set of prime numbers contains arbitrarily long arithmetic progressions, since

$\limsup_{N\rightarrow\infty} \dfrac{|P\cap [1,N]|}{\pi(N)}=1$,

where $P$ is the set of all prime numbers.

## Numerical work

These results are existence theorems and do not show how to find the progressions. On January 18, 2007, Jarosław Wróblewski found the first known case of 24 primes in arithmetic progression:[3]

468,395,662,504,823 + 205,619 · 223,092,870 · n, for n = 0 to 23.

The constant 223092870 here is the product of the prime numbers up to 23 (see primorial).

On May 17, 2008, Wróblewski and Raanan Chermoni found the first known case of 25 primes:

6,171,054,912,832,631 + 366,384 · 223,092,870 · n, for n = 0 to 24.

On April 12, 2010, Benoãt Perichon with software by Wróblewski and Geoff Reynolds in a distributed PrimeGrid project found the first known case of 26 primes (sequence A204189 in OEIS):

43,142,746,595,714,191 + 23,681,770 · 223,092,870 · n, for n = 0 to 25.