On a generalization of R. Chapman's "evil determinant"

Let $p$ be an odd prime and $x$ be an indeterminate. Recently, Z.-W. Sun proposed the following conjecture: $$\det {[x+\left(\frac{j-i}{p}\right)]}_{0\le i,j\le \frac{p-1}{2}}=\{\begin{array}{lll}(\frac{2}{p})p{b}_{p}x-a_p& ifp\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4),1& ifp\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}4),\end{array}$$ where $a_p$ and $b_p$ are rational numbers related to the fundamental unit and class number of the real quadratic field $\mathbb{Q}(\sqrt{p})$. In this paper, we confirm the above conjecture of Sun based on Vsemirnov’s decomposition of Chapman’s “evil determinant”.