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

Abstract

We show how to construct discretely ordered principal ideal subrings of Q[x] with various types of prime behaviour. Given any set D consisting of finite strictly increasing sequences (d 1 , d 2 , . . . , d l ) of positive integers such that, for each prime integer p, the set {pZ, d 1 +pZ, . . . , d l +pZ} does not contain all the cosets modulo p, we can stipulate to have, for each (d 1 , . . . , d l ) ∈ D, a cofinal set of progressions (f, f + d 1 , . . . , f + d l ) of prime elements in our PID Rτ . Moreover, we can simultaneously guarantee that each positive prime g ∈ Rτ \ N is either in a prescribed progression as above or there is no other prime h in Rτ such that g − h ∈ Z.The study of various types of finite sequences in prime numbers is a firm and celebrated part of the field of number theory. The twin prime conjecture or the Green-Tao theorem, [5], are just two of many universally known problems in the area. Given a strictly increasing sequence d = (d 1 , d 2 , . . . , d l ) of positive integers such that a, a + d 1 , a + d 2 , . . . , a + d l are prime numbers for infinitely many a ∈ P, it easily follows that, for each prime number p, the set {pZ, d 1 +pZ, d 2 +pZ, . . . , d l +pZ} does not contain all the cosets modulo p. This is a simple necessary condition for the existence of cofinal sequence of (finite) prime progressions with differences prescribed by d.In [4], the authors defined and studied certain discretely ordered subrings R τ of Q[x] where Z[x] ⊆ R τ and τ is a fixed parameter from the ring of profinite integers (see the preliminaries section for definitions). Among other things, they showed that all these subrings are quasi-Euclidean (also known as ω-stage Euclidean) but none is actually Euclidean, cf. [4, Theorems 3.2 and 3.9]. Somewhat more surprisingly, they also proved that many of these subrings are actually PIDs. In particular, there are 2 ω pairwise non-isomorphic PIDs among the subrings of Q[x] according to [4, Proposition 3.4].The aim of this paper is to have a closer look at these discretely ordered PIDs with particular emphasis on the behaviour of finite progressions of primes. We demonstrate a way how to construct a profinite integer τ such that primes are rather sparse in the PID R τ , i.e. for each two distinct primes f, g ∈ R τ \ Z the difference f − g does not belong to Z. On the other hand, our main result, Theorem 4.3, shows in particular that there is a τ ∈ Z such that R τ is a PID and the above mentioned simple necessary condition for the existence of a cofinal sequence of prime progressions in R τ is actually sufficient for each d. All our results are rather selfcontained and elementary and should be therefore accessible to a wide mathematical audience.