Definitions

Green–Tao theorem

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

In 2006, Tao and Tamar Ziegler extended the result to cover polynomial progressions. More precisely, given any integer-valued polynomials P1,..., Pk in one unknown m with vanishing constant terms, 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.

These results were existence theorems and did not show how to find the progressions. On January 18, 2007, Jaroslaw Wroblewski found the first known case of 24 primes in arithmetic progression:

468395662504823 + 205619 × 23# × n, for n = 0 to 23 (23# = 223092870).
On May 17, 2008, Wroblewski and Raanan Chermoni found the first known case of 25 primes:
6171054912832631 + 366384 × 23# × n, for n = 0 to 24.