Two Legendre-symbol determinants of Sun and Bernoulli numbers

Source abstract

Let $p>5$ be an odd prime, put $n=(p-1)/2$, and let $\left(\frac{\cdot}{p}\right)$ be the Legendre symbol. Define $D_p^{\pm}$ to be the following determinants $$\begin{aligned} D_p^{\pm}&:=\det\left[(i\pm j)\left(\frac{i\pm j}{p}\right) \right]_{1\leq i,j\leq n}. \end{aligned}$$ 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 $D_p^{\pm} \equiv 0 \pmod p$. This gives a negative answer to a question of the Sun.