On some determinants arising from quadratic residues
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916210483920896 |
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| author | Ren, Chen-Kai Sun, Zhi-Wei |
| author_facet | Ren, Chen-Kai Sun, Zhi-Wei |
| contents | Let $p>3$ be a prime, and let $d\in\mathbb Z$ with $p\nmid d$. For the determinants $$S_m(d,p)=\det\left[(i^2+dj^2)^{m}\right]_{1\leqslant i,j \leqslant (p-1)/2}\ \ \left(\frac{p-1}2\leqslant m\leqslant p-1\right),$$ Sun recently determined $S_m(d,p)$ modulo $p$ when $m\in\{p-2,p-3\}$ and $(\frac {-d}p)=-1$. In this paper, we obtain $S_{p-2}(d,p)$ modulo $p$ in the remaining case $(\frac{-d}p)=1$, and determine the Legendre symbols $(\frac{S_{p-3}\,(d,p)}p)$ and $(\frac{S_{p-4}\,(d,p)}p)$ in some special cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_11547 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On some determinants arising from quadratic residues Ren, Chen-Kai Sun, Zhi-Wei Number Theory 11A15, 11C20, 15A15 Let $p>3$ be a prime, and let $d\in\mathbb Z$ with $p\nmid d$. For the determinants $$S_m(d,p)=\det\left[(i^2+dj^2)^{m}\right]_{1\leqslant i,j \leqslant (p-1)/2}\ \ \left(\frac{p-1}2\leqslant m\leqslant p-1\right),$$ Sun recently determined $S_m(d,p)$ modulo $p$ when $m\in\{p-2,p-3\}$ and $(\frac {-d}p)=-1$. In this paper, we obtain $S_{p-2}(d,p)$ modulo $p$ in the remaining case $(\frac{-d}p)=1$, and determine the Legendre symbols $(\frac{S_{p-3}\,(d,p)}p)$ and $(\frac{S_{p-4}\,(d,p)}p)$ in some special cases. |
| title | On some determinants arising from quadratic residues |
| topic | Number Theory 11A15, 11C20, 15A15 |
| url | https://arxiv.org/abs/2404.11547 |