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

TL;DR

This paper constructs specific discretely ordered principal ideal subrings of $Q[x]$ with controlled prime sequence distributions, demonstrating their non-Euclidean nature and embedding into profinite integers.

Contribution

It introduces a method to engineer prime distributions in principal ideal subrings of $Q[x]$, controlling prime progressions and their properties.

Findings

Constructed subrings with prescribed prime progression behaviors

Guaranteed prime elements either in specified progressions or isolated

All constructed rings are non-Euclidean and embed into profinite integers

Abstract

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.