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In the case of $\Bbb N$ and $\Bbb Z$ the gap between two consecutive primes could be defined roughly speaking as the absolute value of the (1-dimensional) distance between those mentioned consecutive prime numbers, but is there an equivalent for the Gaussian primes in the complex plane?

For instance, this is just an example, I was thinking about a distance defined by the polar coordinates, in two variables, $(r,\alpha)$, being $r$ the Euclidean norm of the complex number and $\alpha$ the angle between the line that joins $0+0i$ and the Gaussian prime $a+bi$ and the x-axis ($\Bbb R$). A smaller angle would be a smaller number, so in the case of two Gaussian primes with the same distance to $0+0i$ (belonging to a same circunference of complex numbers centered on $0+0i$) the smallest one would be the closest prime to $\Bbb R$ (in some cases it would be a prime $\in \Bbb R$ when possible according to the definition of the Gaussian primes). So it would be a two-dimensional distance (probably it would be possible to transform it into a one-dimensional distance).

Regarding the gap between Gaussian primes I found this article: "Bounded gaps between Gaussian primes" but it seems to focus on gaps between two specific primes, but not considering a complete order between all of them. Also it seemed interesting this presentation about if there are infinitely large gaps between Gaussian primes

I would like to ask the following questions:

  1. How is defined the gap between Gaussian primes?

  2. Is there a standard way of defining a total order between Gaussian primes (or in the complex plane $\Bbb C$ in general)?

  3. If somebody could kindly provide some online articles for some orientation on those topics would be very appreciated. Thank you!

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    $\begingroup$ I guess the most natural one would be using $|a+ib|^2$, and if $|a+ib|^2 = |c+id|^2$ using an ordering on $arg(a+ib)$ $\endgroup$ – reuns Apr 3 '16 at 12:08
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    $\begingroup$ You might be interested in this: en.wikipedia.org/wiki/Gaussian_moat $\endgroup$ – Gerry Myerson Apr 3 '16 at 12:13
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    $\begingroup$ @GerryMyerson yes! I was indeed reading about it, thanks! $\endgroup$ – iadvd Apr 3 '16 at 12:19

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