On some determinants with Legendre symbol entries

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 $(\frac{d}{p})=1$, and that $$\left(\frac{W_p}{p}\right)=\{\begin{array}{ll}(-1{)}^{|\{0<k<\frac{p}{4}:(\frac{k}{p})=-1\}|}& \text{if}p\equiv 1(\mathrm{mod}4),\\ (-1{)}^{\lfloor (p+1)/8\rfloor }& \text{if}p\equiv 3(\mathrm{mod}4),\end{array}$$ where $$S(d,p)=\det {\left[\left(\frac{i^2+d{j}^2}{p}\right)\right]}_{1\le i,j\le (p-1)/2}$$ and $$W_p=\det {\left[\left(\frac{i^2-((p-1)/2)!j}{p}\right)\right]}_{0\le i,j\le (p-1)/2}.$$ We also pose some conjectures on determinants, one of which states that $(-1{)}^{\lfloor (p+1)/8\rfloor }{W}_p$ is a square when $p\equiv 3(\mathrm{mod}4)$.

22 pages, final published version

References in corpus (1)

• A remark on Zoloterav's theorem

Cited by in corpus (1)

• Hankel-type determinants for some combinatorial sequences