TITLE: Demonstrate that these two Pell's equations have no integer solutions
QUESTION [5 upvotes]: I would like to demonstrate that the following four Pell's equations have no integer solutions:
$x^2-82y^2=\pm2$
$x^2-82y^2=\pm3$
I do realise that such problems are often solved by algebraic manipulations, reducing modulo prime numbers, and arriving at some contradiction. After some  blind trial and error fumbling in the dark with the above mentioned method, I have decided to consult the community.
All help or input would, as always, be highly appreciated.

REPLY [4 votes]: This is an example of a rather more general phenomenon. The continued fraction expansion of $\sqrt{t^2+1}$ is just $[t,\overline{2t}]$ and hence every convergent $p_i/q_i$ to $\sqrt{t^2+1}$ has the property that
$$
p_i^2- (t^2+1) q_i^2 = \pm 1.
$$
If we have $x^2-(t^2+1)y^2=k$ for a given integer $k$ and some integer $x$ with $y \neq 0$, then
$$
\left| \sqrt{t^2+1} - \frac{x}{y} \right| = \frac{|k|}{y^2 \left|\sqrt{t^2+1} + \frac{x}{y} \right|}
$$
and so $x/y$ is a convergent to $\sqrt{t^2+1}$ provided, roughly, $|k| \leq t$. It follows that the form $x^2-(t^2+1)y^2$ does not represent any non-square integers $k$ with $1< |k| < t$.