论文标题
与某些决定因素有关的legendre符号
Legendre symbols related to certain determinants
论文作者
论文摘要
令$ p $是一个奇怪的素数。对于$ b，c \ in \ mathbb z $，太阳引入了行列$$ d_p（b，c）= \ left |（i^2+bij+cj^2）^{p-2} \ right | _ {1 \ leqslant i，j \ leqslant i，j \ leqslant p-1}，$ $和legendre符号$（\ frac {d_p（b，c）} p）$。最近，她和ni证明了$（\ frac {d_p（1,1）} p）=（\ frac {-2} p）$，如果$ p \ equiv2 \ pmod 3 $，这证实了先前的sunduenture of Sun。在本文中，我们确定$（\ frac {d_p（1,1）} p）$在情况下$ p \ equiv1 \ pmod3 $。太阳证明了$ d_p（2,2）\ equiv0 \ pmod p $如果$ p \ equiv3 \ pmod4 $，相反，我们证明$（\ frac {\ frac {d_p（2,2）} p）= 1 $ $ p \ equiv1 equiv1 equiv1 \ equiv1 \ equiv1 \ pmod8 $（$ pmod8 $），以及$（\ frac $ freac if $ freac if freac if freac） $ p \ equiv5 \ pmod8 $。我们的工具包括通用的三项式系数和卢卡斯序列。
Let $p$ be an odd prime. For $b,c\in\mathbb Z$, Sun introduced the determinant $$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.
