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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:

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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.

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  • $\begingroup$ ($\omega = e^{2 i \pi /3}$) Restrict to the segment $8+n(\omega -1), n \in \{0,\ldots 7\}$, the others being obtained by multiplication with an unit, and you are asking about other segments $a +n(\omega -1), n \in \{0,\ldots a-1\}$ $\endgroup$
    – reuns
    Feb 7, 2017 at 14:50
  • $\begingroup$ @user1952009 That's what I'm actually doing, except that you can even restrict it to ${0,...,\lfloor a/2 \rfloor}$ (because there's reflectional symmetry as well). That doesn't actually speed up the search though, because I'm exiting early on the first prime anyway (so since I haven't found further gaps yet, I won't even reach the end of that range). $\endgroup$ Feb 7, 2017 at 14:54
  • $\begingroup$ The integers on those segment don't seem to have much in common, so I think it is highly non-trivial to expect some theorems about the number of primes on those (maybe as hard as the twin prime conjecture) $\endgroup$
    – reuns
    Feb 7, 2017 at 14:57
  • $\begingroup$ Is there a formal claim that says that they don't have much in common or is this your gut feeling? $\endgroup$
    – flawr
    Feb 7, 2017 at 14:58
  • $\begingroup$ They have in common as much as $n$ with $n+2$, I'd say $\endgroup$
    – reuns
    Feb 7, 2017 at 14:59

1 Answer 1

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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 $k>0$.

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