Bubbles of congruent primes

TL;DR

This paper generalizes Shiu's theorem from integers to imaginary quadratic fields, demonstrating the existence of arbitrarily large clusters of primes all congruent to a modulo q within these fields.

Contribution

It extends the concept of prime bubbles to imaginary quadratic fields, providing a new understanding of prime distribution in algebraic number fields.

Findings

Existence of arbitrarily large prime bubbles in imaginary quadratic fields.

Generalization of Shiu's theorem to algebraic number fields.

Primes in these fields can form long strings congruent to a given residue class.

Abstract

Shiu proved that if a and q are arbitrary coprime integers, then there exist arbitrarily long strings of consecutive primes which are all congruent to a modulo q. We generalize Shiu's theorem to imaginary quadratic fields, where we prove the existence of "bubbles" containing arbitrarily many primes which are all, up to units, congruent to a modulo q.