Do real quadratic fields with unique primary factorization exist? Bumped in Stillwell's book "Elements of Number Theory" into "The real quadratic fields with unique prime factorization are still not known ...". But doesn't $\mathbb{Q}[\sqrt{2}]$'s ring of integers $\mathbb{Z}[\sqrt{2}]$ have that property?

see   https://oeis.org/A003172   and  https://en.wikipedia.org/wiki/List_of_number_fields_with_class_number_one#Real_quadratic_fields
for some detail.
The difference, quite enormous, from imaginary quadratic fields is that those have only a handful of class number one, and it was a huge endeavor to prove the known list complete. https://en.wikipedia.org/wiki/Stark%E2%80%93Heegner_theorem
For real, everyone expects that there are infinitely many. However, in a few cases, notably $$ 229,\; 257, \; 401, $$ despite being primes congruent to $1 \pmod 4,$ the class number is larger than one (although still odd). If the thing gave class number one for all such primes, we would be able to say that we knew there were infinitely many, but it does not always work.
