Integer solutions of Pell equation in bounded regions

Abstract

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 $|x|+|y|\le \hat{I}»$ for any $\hat{I}»>0$. 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 $\hat{I}»$.

13 pages