Factorization in a non-maximal order of a quadratic field (Q1599607)

Let \(K = {\mathbb Q}(\sqrt{d})\) and \(O_f = {\mathbb Z} + {\mathbb Z}f\omega\), where \(\{ 1, \omega \}\) generates the maximal order \(O_K\) (of integers) and \(f\) is a positive integer. Suppose that \(A_f\) consists of those elements of \(O_f\) whose norm is prime to \(f\). A class number \(h_f\) relative to \(O_f\) is defined and conditions on \(f\) for \(h_f = 1\) provided. If \(x \in A_f\) is a nonunit and \(k\) is the order of the relative class group, there is a form of unique factorization of \(x^k\). The results are applied to solving Diophantine equations of the form \(ax^2 + 2bxy - kay^2 = \pm 1\) where \(k = 2, 3, 7\), \(a \geq 3\), \(b \geq 1\) and \(\text{gcd}(a, 2b) = 1\).