The primes contain arbitrarily long polynomial progressions

You need to know: Prime numbers, polynomials, arithmetic progression.

Background: Let ${\mathbb Z}[m]$ denote the set of polynomials in one variable m with integer coefficients.

The Theorem: On 1st October 2006, Terence Tao and Tamar Ziegler submitted to arxiv a paper in which they proved that given any polynomials $P_1,\dots , P_k\in {\mathbb Z}[m]$ with $P_1(0)=\dots=P_k(0)=0$ , there are infinitely many pairs of positive integers $x$ and $m$ such that numbers $x+P_1(m),\dots , x+P_k(m)$ are simultaneously prime.

Short context: The sequence $x+P_1(m),\dots , x+P_k(m)$ is called a polynomial progression. In an earlier work, Green and Tao proved that primes contains arbitrary long arithmetic progressions, which corresponds to the case $P_j(m)=(j-1)m, \, j=1,\dots,k$ . The Theorem generalises this result to polynomial progressions. Moreover, the authors proved that for any $\epsilon>0$ , there are infinitely many pairs $(x,m)$ as in the Theorem with $1 \leq m \leq x^{\epsilon}$ . In addition, they proved that even any positive proportion of primes contains infinitely many polynomial progressions.