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.
, there exist arithmetic progressions of primes with
The problem can be traced back to investigations of Lagrange and Waring from around 1770.
is a subset of the prime numbers such that then for all positive integers
In particular, the entire set of prime numbers contains arbitrarily long arithmetic progressions.
In their later work on the generalized Hardy–Littlewood conjecture, Green and Tao stated and conditionally proved the asymptotic formula for the number of k tuples of primes
[4] Green and Tao's proof has three main components: Numerous simplifications to the argument in the original paper[1] have been found.
Conlon, Fox & Zhao (2014) provide a modern exposition of the proof.
The proof of the Green–Tao theorem does not show how to find the arithmetic progressions of primes; it merely proves they exist.
There has been separate computational work to find large arithmetic progressions in the primes.
The Green–Tao paper states 'At the time of writing the longest known arithmetic progression of primes is of length 23, and was found in 2004 by Markus Frind, Paul Underwood, and Paul Jobling: 56211383760397 + 44546738095860 · k; k = 0, 1, .
On January 18, 2007, Jarosław Wróblewski found the first known case of 24 primes in arithmetic progression:[6] The constant 223,092,870 here is the product of the prime numbers up to 23, more compactly written 23# in primorial notation.
On May 17, 2008, Wróblewski and Raanan Chermoni found the first known case of 25 primes: On April 12, 2010, Benoît Perichon with software by Wróblewski and Geoff Reynolds in a distributed PrimeGrid project found the first known case of 26 primes (sequence A204189 in the OEIS): In September 2019 Rob Gahan and PrimeGrid found the first known case of 27 primes (sequence A327760 in the OEIS): Many of the extensions of Szemerédi's theorem hold for the primes as well.
Independently, Tao and Ziegler[7] and Cook, Magyar, and Titichetrakun[8][9] derived a multidimensional generalization of the Green–Tao theorem.
[10] In 2006, Tao and Ziegler extended the Green–Tao theorem to cover polynomial progressions.
implies the previous result that there arithmetic progressions of primes of length
Tao proved an analogue of the Green–Tao theorem for the Gaussian primes.