On some determinants with Legendre symbol entries

TL;DR

This paper investigates determinants with Legendre symbol entries, establishing new properties and formulas related to their values, and proposes conjectures about their algebraic nature, especially regarding when certain determinants are perfect squares.

Contribution

The paper introduces new results on determinants involving Legendre symbols and formulates conjectures about their algebraic properties, extending understanding of these mathematical objects.

Findings

Proved that (−S(d,p)/p)=1 for certain d and p.

Derived explicit formulas for (W_p/p) depending on p mod 4.

Posed conjecture that a specific determinant is a perfect square when p ≡ 3 mod 4.

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 $$S(d,p)=\det\left[\left(\frac{i^2+dj^2}p\right)\right]_{1\le i,j\le(p-1)/2}$$ 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 $p\equiv 3\pmod4$.