Integer solutions of Pell equation in bounded regions

The Pell equation $x^2-D{y}^2=1$ with non-square $D>1$ has infinitely many integer solutions, yet most research has centered on the asymptotic behavior of fundamental units as $D$ varies. By contrast, the exact distribution of solutions for a fixed $D$ within bounded regions has received little attention. In this paper, we contribute to this direction by giving an explicit enumeration of all solutions to the Pell equation inside the square $\mathrm{\mid }x\mathrm{\mid }+\mathrm{\mid }y\mathrm{\mid }\le \lambda$ for any $\lambda >0$. We further extend our results to the shifted Pell equation ${(x-a)}^2-D{(y-b)}^2=1$ for integers $a$ and $b$, obtaining exact counts for sufficiently large $\lambda$.