Some determinants involving quadratic residues modulo primes
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911951540453376 |
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| author | Sun, Zhi-Wei |
| author_facet | Sun, Zhi-Wei |
| contents | In this paper we evaluate several determinants involving quadratic residues modulo primes. For example, for any prime $p>3$ with $p\equiv3\pmod4$ and $a,b\in\mathbb Z$ with $p\nmid ab$, we prove that $$\det\left[ 1+\tanπ\frac{aj^2+bk^2}p \right]_{1\le j,k \le \frac{p-1}2} = \begin{cases}-2^{(p-1)/2}p^{(p-3)/4}&\text{if}\ (\frac{ab}p)=1, \\p^{(p-3)/4}&\text{if}\ (\frac{ab}p)=-1, \end{cases}$$ where $(\frac{\cdot}p)$ denotes the Legendre symbol. We also pose some conjectures for further research. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_14301 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some determinants involving quadratic residues modulo primes Sun, Zhi-Wei Number Theory 11A15, 11C20, 15A15, 33B10 In this paper we evaluate several determinants involving quadratic residues modulo primes. For example, for any prime $p>3$ with $p\equiv3\pmod4$ and $a,b\in\mathbb Z$ with $p\nmid ab$, we prove that $$\det\left[ 1+\tanπ\frac{aj^2+bk^2}p \right]_{1\le j,k \le \frac{p-1}2} = \begin{cases}-2^{(p-1)/2}p^{(p-3)/4}&\text{if}\ (\frac{ab}p)=1, \\p^{(p-3)/4}&\text{if}\ (\frac{ab}p)=-1, \end{cases}$$ where $(\frac{\cdot}p)$ denotes the Legendre symbol. We also pose some conjectures for further research. |
| title | Some determinants involving quadratic residues modulo primes |
| topic | Number Theory 11A15, 11C20, 15A15, 33B10 |
| url | https://arxiv.org/abs/2401.14301 |