As is well-known, $Z[\sqrt{-5}]$ is not a ufd because $6$ has more than one prime factorization in this ring: $6=2\cdot 3$ and $6=(1+\sqrt{-5})(1-\sqrt{-5})$. But both of these prime factorizations have the same number $(2)$ of prime factors...Am I correct that in $Z[\sqrt{-29}], 30=2\cdot 3\cdot 5$ and $30=(1-\sqrt{-29})(1+\sqrt{-29})$ are prime factorizings of $30$ that have different numbers of factors?
Also would $Z[\sqrt{-2309}]$ give as distinct prime factorizations $2310=2\cdot 3 \cdot 5\cdot 7\cdot 11=(1+\sqrt{-2309})(1-\sqrt{-2309})$? ($2309$ is a prime number).
What about $Z[\sqrt{-30029}]$, would that give $30030=2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13=(1+\sqrt{-30029})(1-\sqrt{-30029})$ as distinct prime factorizations?($30029$ is prime)...Does this show the number of primes in distinct prime factorizings be different? I'm worried that the norms will cause some of my "primes" to be nonprimes. Thanks.
It is a classical result of Carlitz (1960) that a number ring has an element with different length factorizations into irreducibles iff it has class number $> 2.\,$ For a proof, and a survey of related results see Half-factorial domains, a survey by Scott T. Chapman and Jim Coykendall.