\[x^2-11y^2=1\]

(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.