Find positive integer solutions to $x^2-19y^2=1$
I know that $(\pm13,\pm3)$ are odd-integer solutions to the generalized Pell: $x^2-19y^2=\color{brown}{-2}$, using this I have to find a solution for the Pell equation.
First Question: Does generalized Pell have either no solutions or infinitely many ? If so then I have to find another pair of solution $(x',y')$ such that $$(\pm13,\pm3)\neq(x',y')$$ $$x\equiv x'\mod r\ \text{and}\ y\equiv y'\mod r$$ then by a theorem If I set $(x'',y'')=\left(\frac{xx'-19yy'}{-2},\frac{-xy'+xy'}{-2}\right)$, then $\left(|x''|,|y''|\right)$ is a positive integer solution to the Pell equation, is that correct ?
$\bf{EDIT}:$ Using Will Jagy's formula I got $(x,y)=(170,39)$ as a solution, but where does that formula come from ?