Solutions of Pell's equation \(v^2- (ka^2+ b^2)w^2= -k\) by the ideals of the quadratic fields \(\mathbb{Q} (\sqrt{ka^2+ b^2})\) (Q5947330)

The author looks at solutions of the Diophantine equation \(v^2- Dw^2 = -k\) where \(D = ka^2 + b^2\) for \(k\in \{2,3,7,11,19,43,67,163\}\) in terms of orders in \(\mathbb{Q}(\sqrt D)\). However, this is a special case of the quite general criterion provided by this reviewer in the same journal [Expo. Math. 19, 55-73 (2001; Zbl 0983.11010)].