Pell's Equation Example

To find solutions of Pell's equation  
  (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

\[3+ \frac{1}{3}=\frac{10}{3}\]



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.

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