On the general solutions of the diophantine equation \(ax^2+ 2bxy- kay^2= \pm 1\) (Q1567442)

Let \(a, b, k\) be positive integers, and \(k\) be one of the integers 2, 3, 7, 11, 19, 43, 67, 163; let \(\delta = ka^2 + b^2\). To the diophantine equation (1) \(ax^2 + 2bxy - kay^2 = \pm 1\) associate the Pellian equation (2) \(v^2 - \delta w^2 = -k\). The author studies the connection between these equations, and proves, for example, that, if \(\delta\) is nonsquare and (2) is solvable, then among pairs \((a, b)\) for which \(\delta = ka^2 + b^2\) with \(a\) odd, there is exactly one pair for which (1) is solvable. He shows how all solutions of (1) can be generated from a single one using the fundamental solution of \(p^2 - \delta q^2 = \pm 1\).