6
$\begingroup$

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:

enter image description here

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.

  • 0
    ($\omega = e^{2 i \pi /3}$) 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 $a +n(\omega -1), n \in \{0,\ldots a-1\}$ – 2017-02-07
  • 0
    @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). – 2017-02-07
  • 0
    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) – 2017-02-07
  • 0
    Is there a formal claim that says that they don't have much in common or is this your gut feeling? – 2017-02-07
  • 0
    They have in common as much as $n$ with $n+2$, I'd say – 2017-02-07
  • 0
    This does not answer my question, but you seem to suggest that this is your gut feeling, is that correct? – 2017-02-07
  • 0
    @flawr This is rigorous, but not deterministic : the density of primes in the upper-right triangle of $\mathbb{Z}[\omega]$ is $\frac{1}{\ln n}$ [as in $\mathbb{N}$](https://en.wikipedia.org/wiki/Prime_number_theorem) so the "probability" that there is no prime in $a +n(\omega -1), n \in \{0,\ldots a-1\}$ tends to $0$ as $a \to \infty$. And this is really the same argument as for the twin-prime and Goldbach conjecture. – 2017-02-07
  • 0
    @user1952009 Ah yes, that's the statistical argument I was thinking of. I just wasn't sure whether the prime number theorem applied to the Eisenstein integers like that. So that would make it very very unlikely that another such gap exists, although of course it's no proof of non-existence. I think if you flesh that out into a couple more sentences that would still be a nice answer until someone comes up with a more definitive solution. – 2017-02-07
  • 0
    This is a heuristic, not rigour. You probably mean $\omega+1$ and that the "probability" that there is no prime in $\{a+n(\omega+1) \mid n \in 0,...,a-1\}$ is $(1-1/\ln(a)) - (1-1/\ln(a-1)) \to 0 $ for $a\to \infty$. – 2017-02-07
  • 0
    @flawr Well the density $1/\ln n$ is really equivalent to the prime number theorem and the isolated pole at $s=1$ of the zeta function $\zeta_{\mathbb{Q}(e^{2i \pi / 3})}(s) = \displaystyle\sum_{(n,m) \in \mathbb{N}^2 - (0,0), m \le n} |n+m (e^{2i \pi / 3}-1)|^{-2s}$ (or something like that). Yes you can use it, assuming independent-ness, for assigning a probability to what we are talking about, something like $(1-\frac{1}{2\ln a})^a$, using that $\ln |a+m (e^{2i \pi / 3}-1)|^2 \sim 2\ln a$ – 2017-02-07
  • 0
    Have you looked at all at the corresponding question for Gaussian integers? – 2018-02-19
  • 0
    @GTonyJacobs Not really. Do you mean rings of fixed (odd) Manhattan distance or fixed Chebyshev distance? – 2018-02-19
  • 0
    Chebyshev is the one I had in mind. I was playing with Gaussian primes recently when this question occurred to me: Will any "row" from, say, $ni$ to $n+ni$ contain no primes? No idea how to approach it, either, but it's interesting to look at the questions next to each other. – 2018-02-19

0 Answers 0