# Symmetric dominance regions surrounding a Gaussian prime

Let $z=a + b i$ be a complex number which is a Gaussian prime, on neither the $x$- nor the $y$-axis. So $a^2+b^2$ is a prime. Construct a region $D(z)$ surrounding $z$ which is the largest orthogonally convex polygon surrounding $z$ that is empty of Gaussian primes. ($D$ for Dominance.) More precisely, define $D(z)$ as the union of rectangles $R$ that (a) strictly include $z$, but (b) are empty of other Gaussian primes in the interior of $R$. Two examples are shown below: $z=1+i$, and $z=4+5i$, where the marked points on the boundary of $D(z)$ are Gaussian primes:

It seems likely that $D(z)$ is not well-defined, based on a comment of François Brunault to an earlier MO question, "Gaussian prime spirals":

it's unknown whether there are infinitely many Gaussian primes of the form $n+i$ with $n \in \mathbb{Z}$.

Nevertheless, I wonder if this question might avoid running into an unsolved problem:

Q. Is there a $z=a + b i$, with $a \neq 0$ and $b \neq 0$, such that $D(z)$ is symmetric about both a vertical line through $z$ and a horizontal line through $z$?

In the examples above, although $D(1+i)$ is a square, it is not symmetric about $1+i$. And $D(4+5 i)$ is symmetric about a horizontal through $z$ but not a vertical.

My guess is that the answer to Q is Yes, but I just haven't found the $z$ that leads to symmetry. My question can be answered by a single example. I am seeking some (minimal) structure to the distribution of the Gaussian primes. Thanks for thoughts/ideas!

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