Evaluation of a determinant involving Legendre symbols

Chen-Kai Ren, Zhi-Wei Sun

Let $p>3$ be a prime, and let $(\frac{\cdot }{p})$ be the Legendre symbol. Let $A_p(x)$ denote the matrix $[x+a_{ij}{]}_{1⩽i,j⩽(p-1)/2}$, where $$ a_{ij}=\begin{cases}…