On a determinant involving linear combinations of Legendre symbols

Keqin Liu, Zhi-Wei Sun, Li-Yuan Wang

Published 2024-08-13Version 1

In this paper, we study a conjecture of the second author on determinants involving Legendre symbols. For example, we prove that for any prime $p\equiv3\pmod4$, where $(\frac{\cdot}p)$ is the Legendre symbol: $$\det\left[x+\left(\frac{i-j}p\right)+\left(\frac ip\right)y+\left(\frac jp\right)z+\left(\frac{ij}p\right)w\right]_{0\le i,j\le(p-3)/2}=x$$