Consider the Pell-type equation $ax^2 - by^2 = c$. When $c = \pm 1$, the following Stormer-type result turns out to hold (see e.g. Theorem 3.2):
Theorem. Let $a,b$ be coprime positive integers whose product $ab$ is not a perfect square. Consider the equation $ax^2 - by^2 = 1$, let $(x_1, y_1)$ be its minimal positive integer solution and $(x,y)$ be any positive integer solution. If every prime divisor of $x$ divides $a$ or if every prime divisor of $y$ divides $b$, then $(x,y) = (x_1, y_1)$ or $(x,y) = (ax_1^3 + 3bx_1y_1^2, by_1^3 + 3ax_1^2y_1)$.
Moreover, if $a=1$ or $b=1$, then the second option for $(x,y)$ is not possible.
The above paper (see Section 2), as well as many of its references, also contains results for $c = \pm 2, \pm 4$ and that's it. A natural question follows - is there a Stormer type result for any integer $c$? I understand it might be harder to derive such, because for a generic $c$ the set of solutions to a Pell-type equation consists of several chains rather than a single one as in $x^2 - Dy^2 = 1$. Anyway, I still hope something meaningful can be said in this setting.
