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

TL;DR

This paper investigates the prime divisor structure of quadratic polynomial values related to class groups of quadratic number fields, extending previous results and exploring new cases for both imaginary and real quadratic fields.

Contribution

It fills gaps in the classification of integers based on prime divisors of quadratic polynomial values, linking to class number problems in quadratic fields.

Findings

Characterized integers with limited prime divisors of quadratic polynomial values

Extended results to even positive integers and different polynomial forms

Connected polynomial value properties to class number one conditions

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 ${\mathbb Q}(\sqrt{d})$.