List of prime numbers in imaginary quadratic fields with UFD 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.
 A: 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.
A: OEIS has a list of rational primes that are inert in the intersection of Heegner UFDs: https://oeis.org/A309024
