Legendre symbols related to $D_p(b,1)$
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866929367804805120 |
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| author | Luo, Xin-Qi Xia, Wei |
| author_facet | Luo, Xin-Qi Xia, Wei |
| contents | Let $p$ be an odd prime. For any $b,c\in\mathbb{Z}$, Z.-W. Sun introduced the new-type determinant $$D_p(b,c)=|(i^2+bij+cj^2)^{p-2}|_{1\leqslant i,j\leqslant p-1},$$ and studied its arithmetic properties. In this paper we mainly prove that $$\left(\frac{D_p(b,1)}{p}\right)=\left(\frac{2b}{p}\right)$$ when $(\frac{b^2-4}{p})=-1$ and $p\equiv1\pmod 4$. As an application of our result, we confirm several conjectures of Sun. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_19728 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Legendre symbols related to $D_p(b,1)$ Luo, Xin-Qi Xia, Wei Number Theory Let $p$ be an odd prime. For any $b,c\in\mathbb{Z}$, Z.-W. Sun introduced the new-type determinant $$D_p(b,c)=|(i^2+bij+cj^2)^{p-2}|_{1\leqslant i,j\leqslant p-1},$$ and studied its arithmetic properties. In this paper we mainly prove that $$\left(\frac{D_p(b,1)}{p}\right)=\left(\frac{2b}{p}\right)$$ when $(\frac{b^2-4}{p})=-1$ and $p\equiv1\pmod 4$. As an application of our result, we confirm several conjectures of Sun. |
| title | Legendre symbols related to $D_p(b,1)$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2405.19728 |