 If formula_3 is a subset of the prime numbers such that

then for all positive integers formula_5, the set formula_3 contains infinitely many arithmetic progressions of length formula_5.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 formula_9 in arithmetic progression.