  We show how to construct discretely ordered principal ideal subrings of
$\mathbb Q[x]$ with various types of prime behaviour. Given any set $\mathcal
D$ consisting of finite strictly increasing sequences $(d_1,d_2,\dots, d_l)$ of
positive integers such that, for each prime integer $p$, the set $\{p\mathbb Z,
d_1+p\mathbb Z,\dots, d_l+p\mathbb Z\}$ does not contain all the cosets modulo
$p$, we can stipulate to have, for each $(d_1,\dots, d_l)\in \mathcal D$, a
cofinal set of progressions $(f, f+d_1, \dots, f+d_l)$ of prime elements in our
principal ideal domain $R_\tau$. Moreover, we can simultaneously guarantee that
each positive prime $g\in R_\tau\setminus\mathbb N$ is either in a prescribed
progression as above or there is no other prime $h$ in $R_\tau$ such that
$g-h\in\mathbb Z$. Finally, all the principal ideal domains we thus construct
are non-Euclidean and isomorphic to subrings of the ring $\hat{\mathbb{Z}}$ of
profinite integers.
