I know that the matter is settled for $d<0$ and an open problem for $d>0$ but I am asking about already known values. The below theorems are the motivation for asking this question.
Let $d$ be a square-free integer such that $\mathbb{Z}[\sqrt d]$ is a PID and $q$ be an odd prime such that $$(d \mid q) =1 $$ Then there exist integers $u,v$ such that $$q = \mid u^2 - dv^2 \mid $$
and
Let $d = 1 \pmod 4$ be a squarefree integer such that $\mathbb{Z}\left[ \frac{1 + \sqrt d}{2}\right]$ is a PID and $q$ be an odd prime such that $(d \mid q)=1$ then there exist integers $u,v$ such that $$ q = \left| \left(u + \frac v2 \right)^2 - \frac{mv^2}{4} \right| $$
where $(. \mid .)$ is the Legendre's symbol.
So knowing the values of $d$ for which $O_K$ is a PID will be a great advantage in representing primes in quadratic forms. So here I list some of the values that I know.
- This answer lists all the values for
$d<0$ and they are $d=-1, -2 , -3 , -7, -11, -19, -43, -67,$ or $-163$. - In the book Introductory Algebraic number theory by S.Alaca , K.Williams, we can see examples of norm Euclidean domains (so PID) and they are for $d = 2,3,5,6,7,11,13,17,19,21,29,33,37,41,57,$ or $ 73$.
- In the same book we also see that the authors mentioning that for $d = 14,69$ are also Euclidean hence PID.
Are there any other values of $d$ that are known in the literature? It will be great if you can add some references along with your answer at least.
