Legendre symbols related to certain determinants

TL;DR

This paper investigates the Legendre symbols of determinants defined by quadratic forms, extending previous results and providing new evaluations for specific cases using advanced number theory tools.

Contribution

It determines the Legendre symbol $(rac{D_p(1,1)}{p})$ for primes $p ot ot 3$, and characterizes $(rac{D_p(2,2)}{p})$ based on congruences, using generalized trinomial coefficients and Lucas sequences.

Findings

$(rac{D_p(1,1)}{p})$ is determined for $p ot ot 3$

$(rac{D_p(2,2)}{p})$ equals 1 or 0 depending on $p mod 8$

New methods involve generalized trinomial coefficients and Lucas sequences.

Abstract

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.