论文标题
关于Legendre符号的决定因素
Determinants concerning Legendre symbols
论文作者
论文摘要
具有Legendre符号条目的决定因素的评估与有限字段的字符总和密切相关。最近，太阳对这个话题提出了一些猜想。在本文中，我们证明了太阳的一些猜想，还研究了一些变体。例如，我们显示以下结果： 令$ p = a^2+4b^2 $是$ a，b $ integers和$ a \ equiv1 \ pmod4 $的素数。然后，对于确定性$$ s（1，p）：= {\ rm det} \ bigg [\ left（\ frac {\ frac {i^2+j^2} {p} {p} \ right）\ bigg] _ {1 \ le i，j \ le i，j \ le \ le \ le \ frac {p-1} Cohen，Sun和Vsemirnov提出的猜想。
The evaluations of determinants with Legendre symbol entries have close relation with character sums over finite fields. Recently, Sun posed some conjectures on this topic. In this paper, we prove some conjectures of Sun and also study some variants. For example, we show the following result: Let $p=a^2+4b^2$ be a prime with $a,b$ integers and $a\equiv1\pmod4$. Then for the determinant $$S(1,p):={\rm det}\bigg[\left(\frac{i^2+j^2}{p}\right)\bigg]_{1\le i,j\le \frac{p-1}{2}},$$ the number $S(1,p)/a$ is an integral square, which confirms a conjecture posed by Cohen, Sun and Vsemirnov.
