  Let $\mathfrak{p}=(\mathfrak{p}_1,...,\mathfrak{p}_r)$ be a system of $r$
polynomials with integer coefficients of degree $d$ in $n$ variables
$\mathbf{x}=(x_1,...,x_n)$. For a given $r$-tuple of integers, say
$\mathbf{s}$, a general local to global type statement is shown via classical
Hardy-Littlewood type methods which provides sufficient conditions for the
solubility of $\mathfrak{p}(\mathbf{x})=\mathbf{s}$ under the condition that
each of the $x_i$'s is prime.
