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Does anyone have a larger list of the first unique gaussian primes ordered by their norm? Uniqueness can be achieved if one starts with $p_1=1+i$ and requires that for odd primes $p_i \equiv 1 \mod 2+2i$. The first primes are then \begin{align} p_2 &= -1 + 2i \\ p_3 &= -1 - 2i \\ p_4 &= -3 \\ p_5 &= 3 + 2i \\ p_6 &= 3 - 2i \\ p_7 &= 1 + 4i \\ p_8 &= 1 - 4i \\ \vdots \end{align}

Also: Is there something like a prime-number theorem for gaussian primes?

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    $\begingroup$ Those seem to me to be two rather unrelated questions that you should ask separately. $\endgroup$ – joriki Aug 15 '18 at 7:19
  • $\begingroup$ "unique" is confusing in this context : $1+4i$ and $1-4i$ have the same norm. In which sense are they unique then ? $\endgroup$ – Peter Aug 15 '18 at 11:35
  • $\begingroup$ I don't think I want to open a question with just 1 sentence, as people seem to not like it, as they want more context. @Peter: They are unique in the sense the same way as conventional prime numbers are unique: Every gaussian integer can be decomposed uniquely into gaussian primes. $\endgroup$ – Diger Aug 15 '18 at 13:00
  • $\begingroup$ @Diger I still do not get what you mean : Unique upto conjugation and units ? If so, why not just list $1+4i$ ? And why is $4+i$ ruled out ? $\endgroup$ – Peter Aug 15 '18 at 13:04
  • $\begingroup$ Because $4+i \equiv i \mod 2+2i$ and not $1$. It can be represented by multiplying $1-4i$ with the unit $i$. $\endgroup$ – Diger Aug 15 '18 at 13:16

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