## Pell's Equation Example

To find solutions of Pell's equation
$x^2-11y^2=1$
(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.