In number theory, the Green–Tao theorem, proved by Ben Green and Terence Tao in 2004, 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. 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.
Let denote the number of primes less than or equal to . If is a subset of the prime numbers such that
then for all positive integers , the set contains infinitely many arithmetic progressions of length .
This statement implies that the set of prime numbers contains arbitrarily long arithmetic progressions, since
where is the set of all prime numbers.
- 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.
- 43,142,746,595,714,191 + 23,681,770 · 223,092,870 · n, for n = 0 to 25.
- Erdős conjecture on arithmetic progressions
- Dirichlet's theorem on arithmetic progressions
- Arithmetic combinatorics
- Green, Ben; Tao, Terence (2008), "The primes contain arbitrarily long arithmetic progressions", Annals of Mathematics 167 (2): 481–547, arXiv:math.NT/0404188, doi:10.4007/annals.2008.167.481.
- Tao, Terence; Ziegler, Tamar (2008), "The primes contain arbitrarily long polynomial progressions", Acta Mathematica 201: 213–305, arXiv:math.NT/0610050, doi:10.1007/s11511-008-0032-5.
- Jens Kruse Andersen, Primes in Arithmetic Progression Records. Retrieved on 2014-06-13
- MathWorld news article on proof
- Primes in Arithmetic Progression Records
- P. Erdos and P.Turán, On some sequences of integers, J. London Math. Soc. 11 (1936), 261–264.
- AMS lecture: Structure and randomness in the prime numbers  by Terence Tao.
- The Green-Tao theorem: an exposition by David Conlon, Jacob Fox, and Yufei Zhao
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