On some conjectural determinants of Sun involving residues
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913423404564480 |
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| author | Chaliha, Rituparna Kalita, Gautam |
| author_facet | Chaliha, Rituparna Kalita, Gautam |
| contents | For an odd prime $p$ and integers $d, k, m$ with gcd$(p,d)=1$ and $2\leq k\leq \frac{p-1}{2}$, we consider the determinant
\begin{equation*}
S_{m,k}(d,p) = \left|(α_i - α_j)^m\right|_{1 \leq i,j \leq \frac{p-1}{k}},
\end{equation*}
where $α_i$ are distinct $k$-th power residues modulo $p$. In this paper, we deduce some residue properties for the determinant $S_{m,k}(d,p)$ as a generalization of certain results of Sun. Using these, we further prove some conjectures of Sun related to $$\left(\frac{\sqrt{S_{1+\frac{p-1}{2},2}(-1,p)}}{p}\right) \text{ and } \left(\frac{\sqrt{S_{3+\frac{p-1}{2},2}(-1,p)}}{p}\right).$$ In addition, we investigate the number of primes $p$ such that $p\ |\ S_{m+\frac{p-1}{k},k}(-1,p)$, and confirm another conjecture of Sun related to $S_{m+\frac{p-1}{2},2}(-1,p)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_07085 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On some conjectural determinants of Sun involving residues Chaliha, Rituparna Kalita, Gautam Number Theory 11A07, 11A15, 11C20, 15A15, 15B36, 11R11 For an odd prime $p$ and integers $d, k, m$ with gcd$(p,d)=1$ and $2\leq k\leq \frac{p-1}{2}$, we consider the determinant \begin{equation*} S_{m,k}(d,p) = \left|(α_i - α_j)^m\right|_{1 \leq i,j \leq \frac{p-1}{k}}, \end{equation*} where $α_i$ are distinct $k$-th power residues modulo $p$. In this paper, we deduce some residue properties for the determinant $S_{m,k}(d,p)$ as a generalization of certain results of Sun. Using these, we further prove some conjectures of Sun related to $$\left(\frac{\sqrt{S_{1+\frac{p-1}{2},2}(-1,p)}}{p}\right) \text{ and } \left(\frac{\sqrt{S_{3+\frac{p-1}{2},2}(-1,p)}}{p}\right).$$ In addition, we investigate the number of primes $p$ such that $p\ |\ S_{m+\frac{p-1}{k},k}(-1,p)$, and confirm another conjecture of Sun related to $S_{m+\frac{p-1}{2},2}(-1,p)$. |
| title | On some conjectural determinants of Sun involving residues |
| topic | Number Theory 11A07, 11A15, 11C20, 15A15, 15B36, 11R11 |
| url | https://arxiv.org/abs/2407.07085 |