On some determinants and permanents

In this paper we study some determinants and permanents. In particular, we investigate the new type determinants $$\det [(i^2+cij+d{j}^2{)}^{p-2}{]}_{1\le i,j\le p-1}\text{ and }\det [(i^2+cij+d{j}^2{)}^{p-2}{]}_{0\le i,j\le p-1}$$ modulo an odd prime $p$, where $c$ and $d$ are integers. We also pose some conjectures for further research.