The Diophantine equation \(ax^{2}+2bxy-4ay^{2}=\pm 1\) (Q1415077)

The solution of the Diophantine equation \[ ax^2 + 2bxy - 4ay^2 = \pm 1 \tag{1} \] with \(a\) and \(b\) positive integer parameters, \(a > 1\) and gcd\((a, 2b) = 1\), is related to the solution of the Pellian equation \[ v^2 - \delta w^2 = -4 \tag{2} \] where \(\delta = 4a^2 + b^2\). Given a solution \((x, y)\) of (1), \(v = \epsilon (bx^2 - 8axy - 4by^2)\), \(w = x^2 + 4y^2\) gives a solution of (2), where \(\epsilon\) is the sign of \(ax^2 + 2bxy - 4ay^2\). Conversely, if \((v, w)\) is a solution of (2) with \(b + 2ai\) a divisor of one of \(v + 2i\), \(v - 2i\), then the first is solvable. Indeed, given \(\delta\) and a solution \((v, w)\) of (2), there is exactly one odd coprime pair \((a, b)\) for which \(\delta = 4a^2 + b^2\) and (1) is solvable.