Are there further gaps in the Eisenstein primes? I recently played around with Eisenstein primes a bit (in an admittedly very amateurish way) and noticed among other things that there are no primes on the hexagonal ring that goes through (8,0) on the Eisenstein grid of the complex plane:

I thought this was a neat feature of the distribution of the primes and started looking for further such gaps. To my astonishment I haven't been able to find a single such gap up to at least a "radius" of 40,000,000. So now I'm wondering whether 8 is indeed the only such gap (ignoring the trivial cases of 0 and 1), or whether there might be further gaps at larger radii.
My Google efforts haven't turned up anything on this and I'm not sure how one would go about answering the question short of keeping the search running in hopes of finding another gap (which of course will never yield the answer "no further gaps exist"). I assume one could make a statistical argument based on the density of the Eisenstein primes, but I'm not sure how the prime number theorem applies to them.
 A: It seems to me that these problems (both in case of Eisenstein and Gaussian primes) are really hard and outside of today's possibilities. I've checked all possible squares (in case of Gaussian primes) and hexagons (in case of Eisenstein primes) of a size up to $10^9$ and the only "primeless" polygon was the hexagon mentioned by OP and the one eight times smaller. However the proof in case of the Gaussian primes would be equivalent to showing that for every $n$ there exists $0 \le k \le n$ such that $n+ki$ is a gaussian prime which is (almost) equivalent of proving that $n^2+k^2$ is a prime number. It is even unclear why such a $k$ should exist and proving that it should be of a size $O(n)$ seems to be even harder task. Eisenstein primes have similar problems but now (at least for hexagons passing through even integers) the problem is (almost) equivalent to finding $0 \le k \le n$ such that $3n^2+k^2$ is a prime number. I am saying almost because for $k=0$ the criteria are different but it seems to not matter, as one can still find primes with $k>0$. 
