Two Legendre-symbol determinants of Sun and Bernoulli numbers

Let $p>5$ be an odd prime, put $n=(p-1)\mathrm{/}2$, and let $\left(\frac{\cdot }{p}\right)$ be the Legendre symbol. Define $D_p^{\pm }$ to be the following determinants $$\begin{array}{rl}{\displaystyle D_p^{\pm }}& {\displaystyle :=\det {[(i\pm j)\left(\frac{i\pm j}{p}\right)]}_{1\le i,j\le n}\mathrm{.}}\end{array}$$ In this article, we give explicit formulas for these determinants modulo $p$ in terms of Bernoulli numbers. As a consequence of Reinhart’s recent counterexample to the Ankeny–Artin–Chowla conjecture, we find a prime $p$ such that $D_p^{\pm }\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}p)$. This gives a negative answer to a question of the Sun.