 In 2006, Tao and Ziegler extended the Green–Tao theorem to cover polynomial progressions.More precisely, given any integer-valued polynomials "P"1..., "P""k" in one unknown "m" all with constant term 0, there are infinitely many integers "x", "m" such that "x" + "P"1("m"), ..., "x" + "P""k"("m") are simultaneously prime.The special case when the polynomials are "m", 2"m", ..., "km" implies the previous result that there are length "k" arithmetic progressions of primes.