Concept
In number theory, primes in arithmetic progression are any sequence of at least three prime numbers that are consecutive terms in an arithmetic progression. An example is the sequence of primes (3, 7, 11), which is given by for .
According to the Green–Tao theorem, there exist arbitrarily long sequences of primes in arithmetic progression. Sometimes the phrase may also be used about primes which belong to an arithmetic progression which also contains composite numbers. For example, it can be used about primes in an arithmetic progression of the form , where a and b are coprime which according to Dirichlet's theorem on arithmetic progressions contains infinitely many primes, along with infinitely many composites.
For integer k ≥ 3, an AP-k (also called PAP-k) is any sequence of k primes in arithmetic progression. An AP-k can be written as k primes of the form a·n + b, for fixed integers a (called the common difference) and b, and k consecutive integer values of n. An AP-k is usually expressed with n = 0 to k − 1. This can always be achieved by defining b to be the first prime in the arithmetic progression.
Any given arithmetic progression of primes has a finite length. In 2004, Ben J. Green and Terence Tao settled an old conjecture by proving the Green–Tao theorem: The primes contain arbitrarily long arithmetic progressions. It follows immediately that there are infinitely many AP-k for any k.
If an AP-k does not begin with the prime k, then the common difference is a multiple of the primorial k# = 2·3·5·...·j, where j is the largest prime ≤ k.
Proof: Let the AP-k be a·n + b for k consecutive values of n. If a prime p does not divide a, then modular arithmetic says that p will divide every pth term of the arithmetic progression. (From H.J. Weber, Cor.10 in Exceptional Prime Number Twins, Triplets and Multiplets," arXiv:1102.3075[math.NT]. See also Theor.2.3 in Regularities of Twin, Triplet and Multiplet Prime Numbers," arXiv:1103.0447[math.NT], Global J.P.A.Math 8(2012), in press.