Some determinants involving binary forms

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