关于孙的若干涉及留数的行列式猜想
On some conjectural determinants of Sun involving residues
AI总结：
本文研究模素数幂留数构成的行列式$S_{m,k}(d,p)$的留数性质，推广孙的相关结果，证明其两个关于二次剩余符号的猜想，并证实另一相关素数计数类猜想。
摘要：
对于奇素数$p$以及满足$\text{gcd}(p,d)=1$、$2\leq k\leq \frac{p-1}{2}$的整数$d,k,m$，我们研究行列式$S_{m,k}(d,p) = \left|(α_i - α_j)^m\right|_{1 \leq i,j \leq \frac{p-1}{k}}$，其中$α_i$是模$p$的互不相同的$k$次幂留数。本文推导了该行列式的若干留数性质，推广了孙的部分结果。借此我们进一步证明了孙关于$\left(\frac{\sqrt{S_{1+\frac{p-1}{2},2}(-1,p)}}{p}\right)$与$\left(\frac{\sqrt{S_{3+\frac{p-1}{2},2}(-1,p)}}{p}\right)$的若干猜想。此外，我们探究了满足$p\\ |\\ S_{m+\frac{p-1}{k},k}(-1,p)$的素数$p$的数量，并证实了孙关于$S_{m+\frac{p-1}{2},2}(-1,p)$的另一猜想。
英文摘要：
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 $S_{m+\frac{p-1}{2},2}(-1,p)$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
