# Green–Tao theorem

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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.

## Numerical work

Those results were existence theorems and did 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.