TITLE: List of prime numbers in imaginary quadratic fields with UFD
QUESTION [10 upvotes]: I am interested in a list of natural prime numbers in the ring of integers of imaginary quadratic fields with UFD e.g. for $\mathbb{Q}[\sqrt{-7}]$ or $\mathbb{Q}[\sqrt{-11}]$. Especially, I want to know the smallest number that is prime in all of these integer rings that is also prime in $\mathbb{Z}$. The possible values for the discriminant $d$ of $\mathbb{Q}[\sqrt{d}]$ to fulfill these requirements are the Heeger numbers $$-1,-2,-3,-7,-11,-19,-43,-67,-163.$$ 
So far, I only managed to find construction tools for the cases
$$d = -1: p \textrm{ is prime and } p \equiv 3 \textrm{ mod } 4 \, ,$$
$$d = -2: p \textrm{ is prime and } p \equiv \{0,2,4,5,6,7\} \textrm{ mod } 8 \, , $$
$$d = -3: p \textrm{ is prime and } p \equiv 2 \textrm{ mod } 3 \, .$$
I browsed the internet for many hours but could not find restrictions for integer primes for the case $d = -7$ (Kleinian integers). I read that there are no prime elements in the integer ring of Hurwitz quaternions so does this transfer to the Kleinian integers, too ?
Thanks in advance,
Levigeddon.

REPLY [2 votes]: For the most part, the problem is already solved by Robert Soupe.  Here I make some additional observations.  To sum up, we should have expected just under 0.2% of primes to be irreducible over all nine imaginary quadratic UFDs (IQUFDs) -- and that is what we get.

Once the condition $p\equiv-1\bmod 24$ is established, it can be combined with any nonquadratic residue $\bmod 7$, then any nonquadratic residue $\bmod 11$, etc up to $\bmod 163$.  Every such combination produces a unique combination $\bmod(8×3×7×...×163)=\bmod16488700536$.

Numerically this corresponds to  three nonquadratic residues $\bmod7$, times five residues$\bmod 11$, times nine residues $\bmod19$, etc up to 81 residues$\bmod163$, for a total of $7577955$ residues $\bmod16488700536$.  Thus more than "a few" residue classes are allowed when we combine all the relevant moduli.  By Dirichlet's Theorem all such residues allow infinitely many primes, so $3167$ is just the smallest of infinitely many solutions.  If we number the residue classes from $0$ to $16488700535$, then the $3167$ also turns out to be the smallest candidate residue.

The Euler totient function of $16488700536$ is $3879912960=7577955×512$.  Thus $1/512=1/2^9$ of all natural primes are expected to be irreducible over all nine IQUFDs.  This goes along with each domain allowing half the natural primes to be irreducible; the halves corresponding to the nine IQUFDs are statistically uncorrelated with each other.

We would have thus expected the first successful prime to lie somewhere around the 512th overall prime.  The latter is $3671$ compared with the actual answer $3167$, so the statistical prediction is "in the ballpark".

Looking out to the first billion numbers, $50847534$ are prime.  Dividing this by $512$ gives (to the nearest whole number) $99312$ versus the actual count (revealed in the comments) of $99192$ irreducible primes over all nine IQUFDs.  The occurrence of such primes is therefore matching well with what we should have been expecting all along.