\(k\)-reduced ideals of orders of real quadratic fields (Q1907753)

Eisenstein sought a priori criteria for determining when \((*)\) \(|x^2- Dy^2|= 4\) has integer solutions where \(\text{gcd}(x, y)= 1\) and \(D\) is a fundamental discriminant. Authors, such as \textit{A. J. Stephens} and \textit{H. C. Williams} [in: Théorie des nombres, C. R. Conf. Int., Québec/Can. 1987, 869-886 (1989; Zbl 0689.10024)], used the infrastructure of a real quadratic field to find an effective algorithm for determining when \((*)\) has solutions. Note that, when \(D\equiv 5\pmod 8\), the existence of solutions to \((*)\) is equivalent to the fundamental unit of the maximal order \({\mathcal O}_D\) in \(\mathbb{Q}(\sqrt D)\) not being in the non-maximal order \({\mathcal O}_{4D}\), of index 2 in \({\mathcal O}_D\). In the paper under review, the author uses homomorphisms between orders, and defines a grading for reduced ideals. With these tools in hand, he gives criteria for the solution of \((*)\) (too technical to state here). Other results are also presented.