Ideal class groups of some quadratic number fields and factorization of values of some quadratic polynomials

We fill the gaps in A. Gica’s determination of all the odd positive integers $d$ for which the number of distinct prime divisors of $f_d(x)=d+x^2$ is less than or equal to $2$ for all the positive and odd integers $x\le \sqrt{d}$. We also determine all the even positive integers $d$ for which the number of distinct prime divisors of $f_d(x)$ is less than or equal to $2$ for all the positive and even integers $x\le \sqrt{d}$. These problems are related to the famous Frobenius-Rabinowitsch’s characterization of the imaginary quadratic number fields $\mathbb{Q}(\sqrt{-d})$ of odd discriminants with class number one in terms of the primality of $f_d(x)\mathrm{/}4$ for all the positive and odd integers $x\le \sqrt{d}$. However, the solution to our problem is much more difficult to come up with. We also begin to address the same problems for the case of $f_d(x)=d-x^2$, in relation with the class groups of the real quadratic number fields $\mathbb{Q}(\sqrt{d})$.