\[x^2-11y^2=1\]
(using this theorem on Solutions of Pell's Equation are Convergents of an Infinite Continued Fraction)find the convergents of the continued fraction \[[ 3 \lt 3, 6 \gt ] = [ 3,3,6,3,6,3,6,... ]\]
.The convergents are
\[\frac{3}{1}\]
\[3+ \frac{1}{3}=\frac{10}{3}\]
\[3+\frac{1}{3+\frac{1}{6}}=\frac{63}{19}\]
\[3+\frac{1}{3+\frac{1}{6+\frac{1}{3}}}=\frac{199}{60}\]
and so on.
The solutions are found from the 2nd, 4th (and even numbered convergents)
\[x_1=10, \; y_1=3, \; x_2=199, \; y_2=60\]
and so on.