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On some determinants with Legendre symbol entries (1308.2900v14)
Published 13 Aug 2013 in math.NT
Abstract: In this paper we mainly focus on some determinants with Legendre symbol entries. Let $p$ be an odd prime and let $(\frac{\cdot}p)$ be the Legendre symbol. We show that $(\frac{-S(d,p)}p)=1$ for any $d\in\mathbb Z$ with $(\frac dp)=1$, and that $$\left(\frac{W_p}p\right)=\begin{cases}(-1){|{0<k<\frac p4:\ (\frac kp)=-1}|}&\text{if}\ p\equiv1\pmod4, \(-1){\lfloor(p+1)/8\rfloor}&\text{if}\ p\equiv3\pmod4,\end{cases}$$ where $$S(d,p)=\det\left[\left(\frac{i2+dj2}p\right)\right]_{1\le i,j\le(p-1)/2}$$ and $$W_p=\det\left[\left(\frac{i2-((p-1)/2)!j}p\right)\right]_{0\le i,j\le(p-1)/2}.$$ We also pose some conjectures on determinants, one of which states that $(-1){\lfloor(p+1)/8\rfloor}W_p$ is a square when $p\equiv 3\pmod4$.