Solutions of x^2-17y^2= +-1
\[x^2-17y^2=-1\]
we can use the continued fraction representing \[\sqrt{17}\]
. This is \[[ 4,8,8,8,... ] = [4 \lt 8 \gt ]\]
.The first convergent of the continued fraction is
\[4= \frac{4}{1}\]
and a solution of the equation is \[x=4, \; y=1\]
.We can find other solutions by considering odd powers of For
\[4+ \sqrt{17}\]
.For
\[n=3\]
and \[(4+ \sqrt{17})^3=268+65 \sqrt{17}\]
.Another solution is
\[x=268, \; y=65\]
.Likewise, solutions of
\[x^2-17y^2=1\]
can be found by considering the even powers. 
