On some determinants arising from quadratic residues

Chen-Kai Ren, Zhi-Wei Sun

Let $p>3$ be a prime, and let $d\in \mathbb{Z}$ with $p\nmid d$. For the determinants $$S_m(d,p)=\det\left[(i^2+dj^2)^{m}\right]_{1\leqslant i,j \leqslant (p-1)/2}\ \ \left(\frac{p-1…