Eisenstein integers and equilateral ideal triangles
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866911807348670464 |
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| author | McShane, Greg |
| author_facet | McShane, Greg |
| contents | We discuss the relationship between Penner's $λ$-length and the norms of Eisenstein integers. This leads to a geometric proof of the fact, attributed to Fermat, that every prime $p$ of the form $3k + 1$ is the norm of an Eisenstein integer that is can be written as $a^2 - ab + b^2$ for some $a,b \in \mathbb{Z}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_14375 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Eisenstein integers and equilateral ideal triangles McShane, Greg Geometric Topology 57K20, 14J50, 11A41, 11E25 We discuss the relationship between Penner's $λ$-length and the norms of Eisenstein integers. This leads to a geometric proof of the fact, attributed to Fermat, that every prime $p$ of the form $3k + 1$ is the norm of an Eisenstein integer that is can be written as $a^2 - ab + b^2$ for some $a,b \in \mathbb{Z}$. |
| title | Eisenstein integers and equilateral ideal triangles |
| topic | Geometric Topology 57K20, 14J50, 11A41, 11E25 |
| url | https://arxiv.org/abs/2403.14375 |