Green–Tao theorem

In mathematics, 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, for any natural number k, there exist k-term arithmetic progressions of primes. The proof is an extension of Szemerédi's theorem.

Further work

In 2006, Tao and Tamar Ziegler extended the result to cover polynomial progressions.[2] More precisely, given any integer-valued polynomials P₁,..., P_k in one unknown m with vanishing constant terms, there are infinitely many integers x, m such that x + P₁(m), ..., x + P_k(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

These 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 Wroblewski and Geoff Reynolds in a distributed PrimeGrid project found the first known case of 26 primes:

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

See also

* Erdős conjecture on arithmetic progressions
* Dirichlet's theorem on arithmetic progressions
* Arithmetic combinatorics

References

1. ^ Green, Ben; Tao, Terence (2008), "The primes contain arbitrarily long arithmetic progressions", Annals of Mathematics 167: 481–547, doi:10.4007/annals.2008.167.481, arXiv:math.NT/0404188 .
2. ^ Tao, Terence; Ziegler, Tamar (2006), The primes contain arbitrarily long polynomial progressions, arXiv:math.NT/0610050 .
3. ^ Jens Kruse Andersen, Primes in Arithmetic Progression Records. Retrieved on 2010-04-13

External links

* MathWorld news article on proof
* Primes in Arithmetic Progression Records

Number Theory