4 concylic points of Gaussian primes Let $Z_{1}, Z_{2}, Z_{3}$ and $Z_{4}$ be $4$ Gaussian primes different from each other.
There are $4$ concyclic points of Gaussian primes with its center at the origin.
There are $4$ concyclic points of Gaussian primes such that $2$ points of them are the reflection ones in x-axis, y-axis, $ x = y$ line and $x = -y$ line.
Except for the above cases, are there 4 concyclic points of Gaussian primes?
 A: Tao proved in "The Gaussian primes contain arbitrarily shaped constellations" that given any finite configuration of lattice points, you can find a (scaled + translated) copy of that set somewhere in the Gaussian primes.  For instance, the theorem predicts that we can find the corners of an axis-aligned square within the Gaussian primes, and indeed it is quite easy to find one: $2+i, 2+5i, 6+i, 6+5i$, and these are certainly concyclic.
But Tao's theorem gives much more than this.  Using the well-known and elementary Brahmagupta-Fibonacci identity, we can construct circles centered at the origin which contain arbitrarily many lattice points, and then apply the above theorem to such point sets.
This means that there exist arbitrarily many concyclic points consisting entirely of Gaussian primes.  So not only are there 4 concyclic points, there are 3 billion (or any larger number you care to imagine) concyclic points!  But it would be computationally infeasible to find them for even modestly large constellations: I would guess that 30 is already too high for a fixed constellation.  On the other hand we could just pick a large circle crossing many lattice points and then look for Gaussian primes on translates of it.  I bet we could probably find hundreds of concyclic primes this way.
Edit: I see that several similar questions have piqued your interest very recently.  Notice that they are essentially all covered by Tao's theorem, except for the existence of equilateral triangles which is false because it is impossible to find three equidistant lattice points (as achille hui pointed out).
