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We show how to construct discretely ordered principal ideal subrings of$\mathbb Q[x]$ with various types of prime behaviour. Given any set $\mathcalD$ consisting of finite strictly increasing sequences $(d_1,d_2,\dots, d_l)$ ofpositive 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 $(d_1,\dots, d_l)\in \mathcal D$, acofinal set of progressions $(f, f+d_1, \dots, f+d_l)$ of prime elements in ourprincipal ideal domain $R_\tau$. Moreover, we can simultaneously guarantee thateach positive prime $g\in R_\tau\setminus\mathbb N$ is either in a prescribedprogression 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 constructare non-Euclidean and isomorphic to subrings of the ring $\hat{\mathbb{Z}}$ ofprofinite integers.

Controlling distribution of prime sequences in discretely ordered principal ideal subrings of $\mathbb Q[x]$