On Euler's magic matrices of sizes $3$ and $8$
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866916016712318976 |
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| author | Müller, Peter |
| author_facet | Müller, Peter |
| contents | A proper Euler's magic matrix is an integer $n\times n$ matrix $M\in\mathbb Z^{n\times n}$ such that $M\cdot M^t=γ\cdot I$ for some nonzero constant $γ$, the sum of the squares of the entries along each of the two main diagonals equals $γ$, and the squares of all entries in $M$ are pairwise distinct. Euler constructed such matrices for $n=4$. In this work, we construct examples for $n=8$ and prove that no such matrix exists for $n=3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_16260 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Euler's magic matrices of sizes $3$ and $8$ Müller, Peter Combinatorics Number Theory 11C20 (Primary) 15B36 (Secondary) A proper Euler's magic matrix is an integer $n\times n$ matrix $M\in\mathbb Z^{n\times n}$ such that $M\cdot M^t=γ\cdot I$ for some nonzero constant $γ$, the sum of the squares of the entries along each of the two main diagonals equals $γ$, and the squares of all entries in $M$ are pairwise distinct. Euler constructed such matrices for $n=4$. In this work, we construct examples for $n=8$ and prove that no such matrix exists for $n=3$. |
| title | On Euler's magic matrices of sizes $3$ and $8$ |
| topic | Combinatorics Number Theory 11C20 (Primary) 15B36 (Secondary) |
| url | https://arxiv.org/abs/2504.16260 |