Intersective polynomials and the primes

Intersective polynomials are polynomials in Z[x] having roots every modulus. For example, P1(n)=n2 and P2(n)=n2−1 are intersective polynomials, but P3(n)=n2+1 is not. The purpose of this note is to deduce, using results of Green and Tao (2006) [8], and Lucier (2006) [16], that for any intersective polynomial h, inside any subset of positive relative density of the primes, we can find distinct primes p1,p2 such that p1−p2=h(n) for some integer n. Such a conclusion also holds in the Chen primes (where by a Chen prime we mean a prime number p such that p+2 is the product of at most 2 primes).