A Pell’s equation has one trivial solution, \((x,y)=(1,0)\),
corresponding to solution \(z=1\) of equation \(N(z)=1\). But if we
know the smallest *non-trivial* solution, then we can derive
all the solutions. This is what the following statement claims.

Note that \(z=x+y\sqrt d\) determines \(x\) and \(y\) by the formulas \(x=\frac{z+\overline{z}}2\) and \(y=\frac{z-\overline{z}}{2\sqrt d}\). Thus all the solutions of the Pell’s equation are given by the formulas \[x=\frac{z_0^n+\overline{z_0}^n}2\quad\mbox{i}\quad y=\frac{z_0^n-\overline{z_0}^n}{2\sqrt d}.\]

Now we will show that every Pell’s equation indeed has a non-trivial solution.