  In this paper, we study arithmetic properties of certain determinants
involving powers of $i^2+cij+dj^2$, where $c$ and $d$ are integers. For
example, for any odd integer $n>1$ with $(\frac dn)=-1$ we prove that $\det [
(\frac{i^2+cij+dj^2}{n})]_{0\le i,j\le n-1}$ is divisible by $\varphi(n)^2$,
where $(\frac{\cdot}{n})$ is the Jacobi symbol and $\varphi$ is Euler's totient
function. This confirms a previous conjecture of the second author.
