Integer solutions of Pell equation in bounded regions
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916960931938304 |
|---|---|
| author | Ong, Kun Yi Ismail, Eddie Shahril Bin |
| author_facet | Ong, Kun Yi Ismail, Eddie Shahril Bin |
| contents | The Pell equation $x^2 - Dy^2 = 1$ 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 $|x| + |y| \leq λ$ for any $λ> 0$. 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 $λ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_17882 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Integer solutions of Pell equation in bounded regions Ong, Kun Yi Ismail, Eddie Shahril Bin Number Theory 11D09, 11D45 The Pell equation $x^2 - Dy^2 = 1$ 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 $|x| + |y| \leq λ$ for any $λ> 0$. 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 $λ$. |
| title | Integer solutions of Pell equation in bounded regions |
| topic | Number Theory 11D09, 11D45 |
| url | https://arxiv.org/abs/2509.17882 |