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

Abstract

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 $x\leq\sqrt{d}$. 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 $x\leq\sqrt{d}$. 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 $f_d(x)/4$ for all the positive and odd integers $x\leq\sqrt{d}$. 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 ${\mathbb Q}(\sqrt{d})$.