How to solve this Pell's equation $x^{2} - 991y^{2} = 1 $ How to solve the following Pell's equation?
$$x^{2} - 991y^{2} = 1 $$
where $(x, y)$ are naturals.
The answer is $$x = 379,516,400,906,811,930,638,014,896,080$$ 
$$y = 12,055,735,790,331,359,447,442,538,767$$
I can't think of any way apart from brute force. Please help.
Also, what is the general way of solving any Pell's equation?
I read the wiki article on it but couldn't get any general method to solve it.

$$x^2 - 991y^2 = 1$$ -------(1)
Assuming $y \not= 0$ divide both sides of the equation by $y^2$
i.e. $$(x/y)^2 - 991 = (1/y)^2$$
i.e. $(x/y)^2 - (1/y)^2 = 991$
i.e. $(x/y - 1/y)(x/y + 1/y) = (1)(991)$
i.e. $(x/y - 1/y)(x/y + 1/y) = (496 - 495)(496 + 495)$
i.e. $x/y = 496$ and $1/y = 495$
i.e. $x = (496/495)$ and $y = 1/495$
The equation represents a hyperbola of the form $$x^2/a^2 - y^2/b^2 = 1$$
i.e. the equation is $$x^2/(1^2) - y^2/(\sqrt {1/991})^2 = 1$$
foci of this hyperbola are the co ordinates $(\pm c, 0)$
Here $c = \sqrt{a^2 + b^2}$
The asymptotes are $y = \pm (b/a)x$
i.e. The asymptotes are $y = \pm (1/\sqrt{991})x$
There exists a $CONJUGATE$  $HYPERBOLA$ of the form $\frac{x^2}{a^2} - \frac{y^2}{b^2} = -1$, for the $HYPERBOLA$ $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$
Here the equation of the $conjugate$  $hyperbola$ is $x^2 - 991(y^2) = -1$
$$x^2 - 991y^2 = -1$$ -------(2)
Now solve equations (1) and (2) for $x$ and $y$
i.e. $(2)(x^2) - (2)(991)(y^2) = 0$
i.e. $(x^2) - (991)(y^2) = 0$
i.e. $(x^2) = (991)(y^2)$
i.e. $\frac{x^2}{y^2} = 991$
i.e. $\left(\frac{x^2}{y^2} - (\sqrt{991})^2\right) = 0$
i.e. $\left(\frac{x}{y} - \sqrt{991}\right)\left(\frac{x}{y} + \sqrt{991}\right) = 0$
i.e. $(\frac{x}{y} = \pm\sqrt{991})$
i.e. $\frac{y}{x} = \pm\frac{1}{\sqrt{991}}$ , the equations of the asymptotes to the hyperbola
