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arXiv · 2405.19728

Legendre symbols related to $D_p(b,1)$

Xin-Qi Luo·Wei Xia
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Abstract

Let $p$ be an odd prime. For any $b,c\in\mathbb{Z}$, Z.-W. Sun introduced the new-type determinant $$D_p(b,c)=|(i^2+bij+cj^2)^{p-2}|_{1\leqslant i,j\leqslant p-1},$$ and studied its arithmetic properties. In this paper we mainly prove that $$\left(\frac{D_p(b,1)}{p}\right)=\left(\frac{2b}{p}\right)$$ when $(\frac{b^2-4}{p})=-1$ and $p\equiv1\pmod 4$. As an application of our result, we confirm several conjectures of Sun.

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Xin-Qi Luo, Wei Xia. 2024-05-30. Legendre symbols related to $D_p(b,1)$. https://arxiv.org/abs/2405.19728

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