Are there integral domains unknown to be UFD? Are there integral domains unknown to be UFD ??
Im not asking about an infinite set of rings , but a specific integral domain unknown to be a UFD.
What are the simplest examples ?
Are there patterns in the norms of these problematic integral domains ?
 A: It is conjectured that the maximal real subfield of the cyclotomic field $\mathbf Q(\zeta_{2^n})$ has class number one, or equivalently is a UFD, for all $n$ but this is only proved for $n\leq 7$. See https://mathoverflow.net/questions/82480/non-trivial-class-number-at-some-finite-level-in-the-cyclotomic-mathbfz-p-e. So taking $n=8$ gives a specific example for the OP's first question at the time this is being written. It is a number field of degree $2^{8-1} = 128$ over $\mathbf Q$. In the future if the case $n=8$ gets settled then change $n$ to $9$. If the conjecture is settled (affirmatively) for all $n$ then I will happily sacrifice the correctness of this answer.
A: For a Dedekind domain, UFD is equivalent to PID. Hence , for the ring of integers $A_k$ of a number field $k$, UFD is equivalent to "  class number 1 ". Among all imaginary quadratic fields $k= \mathbf Q(\sqrt d)$ the only $A_k$ which are PID correspond to $d=-1, -2, -, -7, -11, -19, -43, -67, -163$ (conjecture of Gauss, theorem of Heegner-Stark). The analogous problem for real quadratic fields is open : it is known for example that $A_k$ is a PID for $d=2, 3, 5, 6,7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, 73$, but could there be an infinite number of such PID's ?
Among the cyclotomic fields $\mathbf Q(\zeta_n)$ , the only $A_k$ which are PID correspond to $n$=1, 3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 24, 25, 27, 28, 32, 33, 35, 36, 40, 44, 45, 48, 60, 84, as well as the values  $2n$ of the odd $n$'s in this list (because $\mathbf Q(\zeta_n)=\mathbf Q(\zeta_{2n})$) . 
