Integer solutions of Pell equation in bounded regions

Abstract: 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 $|x| + |y| \leq \lambda$ for any $\lambda > 0$. We further extend our results to the shifted Pell equation $$\left(x-a\right)² - D\left(y-b\right)² = 1$$ for integers $a$ and $b$, obtaining exact counts for sufficiently large $\lambda$.