Integer solutions of Pell equation in bounded regions

The Pell equation x² - Dy² = 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 |x| + |y| ≤ λ for any $λ> 0$. 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 $λ$.