According to the Green–Tao theorem, there exist arbitrarily long arithmetic progressions in the sequence of primes.
Sometimes the phrase may also be used about primes which belong to an arithmetic progression which also contains composite numbers.
For integer k ≥ 3, an AP-k (also called PAP-k) is any sequence of k primes in arithmetic progression.
Green and Terence Tao settled an old conjecture by proving the Green–Tao theorem: The primes contain arbitrarily long arithmetic progressions.
The first to be discovered was found on April 12, 2010, by Benoît Perichon on a PlayStation 3 with software by Jarosław Wróblewski and Geoff Reynolds, ported to the PlayStation 3 by Bryan Little, in a distributed PrimeGrid project:[2] By the time the first AP-26 was found the search was divided into 131,436,182 segments by PrimeGrid[4] and processed by 32/64bit CPUs, Nvidia CUDA GPUs, and Cell microprocessors around the world.
Before that, the record was an AP-25 found by Raanan Chermoni and Jarosław Wróblewski on May 17, 2008:[2] The AP-25 search was divided into segments taking about 3 minutes on Athlon 64 and Wróblewski reported "I think Raanan went through less than 10,000,000 such segments"[5] (this would have taken about 57 cpu years on Athlon 64).
[6] The following table shows the largest known AP-k with the year of discovery and the number of decimal digits in the ending prime.
Note that unlike an AP-k, all the other numbers between the terms of the progression must be composite.
The first known CPAP-10 was found in 1998 by Manfred Toplic in the distributed computing project CP10 which was organized by Harvey Dubner, Tony Forbes, Nik Lygeros, Michel Mizony and Paul Zimmermann.
The requirement for at least 23090 composite numbers between the 11 primes makes it appear extremely hard to find a CPAP-11.
The table shows the largest known case of k consecutive primes in arithmetic progression, for k = 3 to 10. xd is a d-digit number used in one of the above records to ensure a small factor in unusually many of the required composites between the primes.