Arithmetic progressions of gaussian primes Given $u\in\mathbb{C}$ and $v\in\mathbb{C}$ let's consider the following progression:
$$z_n=u+nv\;\;\;\;\;\;\;\;\;n\ge 0$$

Is it possible to find progressions $z_n$ generating
gaussian primes for an arbitrary long sequence of consecutive values
of n?

For example, $z_n=-13-2i+n(3+i)$ generates gaussian primes for all values $0\le n\le 8$ (examine the norm $|z_n|^2=10n^2-82n+173$):


If not, it is known the progression of maximum lenght?

Many thanks.
 A: The Green--Tao theorem also shows that there are arbitrarily long arithmetic progressions among the (rational) primes that are congruent to 3 modulo 4. See for instance this MO question Is the Green-Tao theorem true for primes within a given arithmetic progression?.
Since any rational prime that is 3 mod 4 is a Gaussian prime, this shows that the Gaussian primes contain arbitrarily long arithmetic progressions.
(This is perhaps a slightly unsatisfactory class of examples. I don't know if there are arbitrarily long arithmetic progressions of primes in $\mathbf{Z}[i]$ which aren't in $\mathbf{Z}$ or $i \mathbf{Z}$.)
A: A theorem of Tao arxiv.org/abs/math/0501314 says: given any finite sets of points $v_i \in \mathbb{Z}[i]$
there are infinitely many $a\in \mathbb{Z}[i],r\in \mathbb{Z}\setminus \{0\}$ such that all $a+rv_i$ are Gaussian primes. Choosing a shape of two parallel lines, say $v_{1,j}=j,v_{2,j}=i+j,j\in \{1, \ldots,k\}$, shows that there are also long progressions of Gaussian primes not all on the real line, (which also answers the question left open by David). One could also take lines, say, with 45 degrees angle.
