  Let $E\subset \mathbb Z$ be a set of positive upper density. Suppose that
$P_1,P_2,..., P_k\in \mathbb Z[X]$ are polynomials having zero constant terms.
We show that the set $E\cap (E-P_1(p-1))\cap ... \cap (E-P_k(p-1))$ is
non-empty for some prime number $p$. Furthermore, we prove convergence in $L^2$
of polynomial multiple averages along the primes.
