Generalization of the Pell's equation

The Pell's equation is a Diophantine equation of the form :

x2−Dy2=1

Where Dis a positive integer different from a perfect square.

The trivial solution to a pell equation is the couple (x,y)=(1,0)

Let (x1,y1)be the non-trivial minimum solution to the Pell's equation. Then all the

(xn,yn) are given by :

xn=k=0∑⌊2n​⌋​⎝⎛​n2k​⎠⎞​Dk(x1)n−2k(y1)2k

yn=l=0∑⌊2(n−1)​⌋​⎝⎛​n2l+1​⎠⎞​Dl(x1)n−2l−1(y1)2l+1

APPLICATION