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Keywords:
quadratic field; imaginary quadratic field; class group; class number; quadratic polynomial; Frobenius-Rabinowitsch
quadratic field; imaginary quadratic field; class group; class number; quadratic polynomial; Frobenius-Rabinowitsch
Summary:
We fill the gaps in 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 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 positive and even integers $x\leq \sqrt {d}$. These problems are related to 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 $\frac 14 f_d(x)$ for all 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 real quadratic number fields ${\mathbb Q}(\sqrt {d})$.
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