Xin-Qi Luo, Zhi-Wei Sun
Published 2022-10-26Version 1
Let $p$ be an odd prime. For $b,c\in\mathbb Z$, Sun introduced the determinant $$D_p(b,c)=\left|(i^2+bij+cj^2)^{p-2}\right|_{1\leqslant i,j \leqslant p-1},$$ and investigated the Legendre symbol $(\frac{D_p(b,c)}p)$. Recently Wu, She and Ni proved that $(\frac{D_p(1,1)}p)=(\frac {-2}p)$ if $p\equiv2\pmod 3$, which confirms a previous conjecture of Sun. In this paper we determine $(\frac{D_p(1,1)}p)$ in the case $p\equiv1\pmod3$. Sun proved that $D_p(2,2)\equiv0\pmod p$ if $p\equiv3\pmod4$, in contrast we prove that $(\frac{D_p(2,2)}p)=1$ if $p\equiv1\pmod8$, and $(\frac{D_p(2,2)}p)=0$ if $p\equiv5\pmod8$. Our tools include generalized trinomial coefficients and Lucas sequences.
On determinants involving $(\frac{j+k}p)\pm(\frac{j-k}p)$
On determinants involving $(\frac{j^2-k^2}p)$ and $(\frac{jk}p)$
On a determinant involving linear combinations of Legendre symbols
![QR code for arXiv:2210.14741 [math.NT]](http://oss.arxitics.com/images/favicon.svg)