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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2512.15237 |
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| _version_ | 1866912770818048000 |
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| author | Halbeisen, Lorenz Hungerbühler, Norbert Zargar, Arman Shamsi |
| author_facet | Halbeisen, Lorenz Hungerbühler, Norbert Zargar, Arman Shamsi |
| contents | We consider the problem of finding integer triangles with $R/r$ a positive rational, where $R$ and $r$ are the radii of the circumcircle and an excircle, respectively. We show that for general triangles $R/r>1/4$ applies. The equation $R/r=N$ turns out to be related to the elliptic curve $\mathcal{E}_N$ given by $v^2=u^3+2(2N^2+2N-1)u^2-(4N-1)u$. If $N>1/4$ is rational, then the torsion group of $\mathcal{E}_N$ is $\mathbb Z/2\mathbb Z\times\mathbb Z/6\mathbb Z$ if $N(N+2)$ is a square and $\mathbb Z/6\mathbb Z$ otherwise. We show that a rational triangle with rational ratio $R/r=N$ exists if and only if $N>1/4$ and there exists a rational non-torsion point on the curve $\mathcal{E}_N$ which satisfies a certain condition. Furthermore, we show that the rank of $\mathcal{E}_N$ is positive when $N = m^2 \pm 1>1/4$ for a rational $m$. We also show that on every curve $\mathcal{E}_N$ whose rank is positive, there are infinitely many rational points which lead to infinitely many non-similar integer triangles with $R/r=N$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_15237 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Integer triangles with a rational ratio of circumcircle radius to excircle radius Halbeisen, Lorenz Hungerbühler, Norbert Zargar, Arman Shamsi Number Theory 11D72, 11G05 We consider the problem of finding integer triangles with $R/r$ a positive rational, where $R$ and $r$ are the radii of the circumcircle and an excircle, respectively. We show that for general triangles $R/r>1/4$ applies. The equation $R/r=N$ turns out to be related to the elliptic curve $\mathcal{E}_N$ given by $v^2=u^3+2(2N^2+2N-1)u^2-(4N-1)u$. If $N>1/4$ is rational, then the torsion group of $\mathcal{E}_N$ is $\mathbb Z/2\mathbb Z\times\mathbb Z/6\mathbb Z$ if $N(N+2)$ is a square and $\mathbb Z/6\mathbb Z$ otherwise. We show that a rational triangle with rational ratio $R/r=N$ exists if and only if $N>1/4$ and there exists a rational non-torsion point on the curve $\mathcal{E}_N$ which satisfies a certain condition. Furthermore, we show that the rank of $\mathcal{E}_N$ is positive when $N = m^2 \pm 1>1/4$ for a rational $m$. We also show that on every curve $\mathcal{E}_N$ whose rank is positive, there are infinitely many rational points which lead to infinitely many non-similar integer triangles with $R/r=N$. |
| title | Integer triangles with a rational ratio of circumcircle radius to excircle radius |
| topic | Number Theory 11D72, 11G05 |
| url | https://arxiv.org/abs/2512.15237 |