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Is Dirichlet's theorem on arithmetic progressions true for ring of Gaussian integers $\mathbb{Z}[i]$?

By Dirichlet's theorem I mean the fact that if some line $an+b$ ($a, b\in Q$ and $n\in\mathbb{Z}$ is variable) contains two different primes in the ring $Q$, then there are infinitely many primes on it.

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  • $\begingroup$ Note that $1+ni$ is prime iff $n^2+1$ is prime, so showing that the progression $\{1,1+i,1+2i,\cdots\}$ contains infinitely many primes is equivalent to showing that there are infinitely many primes of the form $n^2+1$, which is currently unknown. $\endgroup$ – lulu Oct 8 '18 at 13:35
  • $\begingroup$ You are right, but I think You mean $\{1+i, 2+i, 3+i,\ldots\}$. $\endgroup$ – solver6 Oct 8 '18 at 13:39
  • $\begingroup$ Either way. Yours has period $1$, mine has period $i$. $\endgroup$ – lulu Oct 8 '18 at 13:40
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    $\begingroup$ Please state what the formulation you want is. There are several ways to describe something that could be considered an analogue of Dirichlet's theorem for the Gaussian integers. For example, there is Hecke's equidistribution theorem for Gaussian primes in an angular sector of the complex plane. Perhaps you mean a version about Gaussian primes $\pi$ lying in a Gaussian arithmetic progression, namely $\pi \equiv \alpha \bmod \mu$ where $\alpha$ and $\mu$ are relatively prime and $\pi$ is suitably normalized to pin down a unit multiple. $\endgroup$ – KCd Oct 8 '18 at 13:43

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