Consider a Pell equation in the form $$X^2-D Y^2=1$$ such that $(a,b)$ is the minimal solution. Now consider the generalized Pell equation $$X^2-DY^2=N^2$$ for some integer $N$: obviously it admits a solution, for example $(Na,Nb)$. My question is: can we say that $(Na,Nb)$ is the minimal solution?
A priori I would answer in a negative way, but can one find at least some conditions on $a,b,D,N$ such that $(Na,Nb)$ is the minimal solution?