Bubbles of congruent primes

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.