Pell's equation solution set

Question

Can we guarantee that pell equation has infinitely many solutions in positive integers without finding a non trivial solution?

Answer 1

If you mean $$ x^2 - n y^2 = 1, $$ with fixed integer $n > 0$ but NOT an integer square, then yes, infinitely many solutions in a Fibonacci type sequence.

It is a non-trivial result that there is at least one solution, take the smallest integers $u,v > 0$ with $u^2 - n v^2 = 1.$ Call $(u,v)$ the fundamental solution.

The automorphism generator matrix is $$ \left( \begin{array}{cc} u & nv \\ v & u \end{array} \right) $$ which means $$ (x_{k+1}, y_{k+1}) = (u x_k + nv y_k \; , \; \; v x_k + u y_k) . $$ The sequences can be separated, $x_0 = 1, y_0 = 0,$ then $x_1 = u, y_1 = v,$ then $$ x_{k+2} = 2u x_{k+1} - x_k \; , $$ $$ y_{k+2} = 2u y_{k+1} - y_k \; , $$