Bounded gaps between Gaussian primes

We show that there are infinitely many distinct rational primes of the form p1=a2+b2 and p2=a2+(b+h)2, with a,b,h integers, such that |h|≤246. We do this by viewing a Gaussian prime c+di as a lattice point (c,d) in R2 and showing that there are infinitely many pairs of distinct Gaussian primes (c1,d1) and (c2,d2) such that the Euclidean distance between them is bounded by 246. Our method, motivated by the work of Maynard [9] and the Polymath project [13], is applicable to the wider setting of imaginary quadratic fields with class number 1 and yields better results than those previously obtained for gaps between primes in the corresponding number rings.