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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | https://arxiv.org/abs/2405.02112 |
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| _version_ | 1866910433869299712 |
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| author | Wang, Li-Yuan Wu, Hai-Liang Ni, He-Xia |
| author_facet | Wang, Li-Yuan Wu, Hai-Liang Ni, He-Xia |
| contents | Let $p$ be an odd prime and $x$ be an indeterminate. Recently, Z.-W. Sun proposed the following conjecture: $$\det\left[x+\left(\frac{j-i}{p}\right)\right]_{0\le i,j\le \frac{p-1}{2}}=\begin{cases} (\frac{2}{p})pb_px-a_p & \mbox{if}\ p\equiv 1\pmod4, 1 & \mbox{if}\ p\equiv 3\pmod4, \end{cases}$$ where $a_p$ and $b_p$ are rational numbers related to the fundamental unit and class number of the real quadratic field $\mathbb{Q}(\sqrt{p})$. In this paper, we confirm the above conjecture of Sun based on Vsemirnov's decomposition of Chapman's "evil determinant". |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_02112 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On a generalization of R. Chapman's "evil determinant" Wang, Li-Yuan Wu, Hai-Liang Ni, He-Xia Number Theory Primary 11C20, Secondary 11L05, 11R18 Let $p$ be an odd prime and $x$ be an indeterminate. Recently, Z.-W. Sun proposed the following conjecture: $$\det\left[x+\left(\frac{j-i}{p}\right)\right]_{0\le i,j\le \frac{p-1}{2}}=\begin{cases} (\frac{2}{p})pb_px-a_p & \mbox{if}\ p\equiv 1\pmod4, 1 & \mbox{if}\ p\equiv 3\pmod4, \end{cases}$$ where $a_p$ and $b_p$ are rational numbers related to the fundamental unit and class number of the real quadratic field $\mathbb{Q}(\sqrt{p})$. In this paper, we confirm the above conjecture of Sun based on Vsemirnov's decomposition of Chapman's "evil determinant". |
| title | On a generalization of R. Chapman's "evil determinant" |
| topic | Number Theory Primary 11C20, Secondary 11L05, 11R18 |
| url | https://arxiv.org/abs/2405.02112 |