A Determinant Congruence Conjectured by Sun

We prove a strengthened form of a conjecture of Sun on a determinant attached to a binary quadratic form. Let $n>3$ and let $c,d\in \mathbb{Z}$. If $n$ is composite, then $$\det [(i^2+cij+d{j}^2{)}^{n-2}{]}_{0\le i,j\le n-1}\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}{n}^2)$$ with no condition on $c$ and $d$. If $n=p$ is prime, the same congruence holds whenever the Legendre symbol $\text{leg}dp$ is $-1$. For composite $n$, a polynomial determinant is divisible by two Vandermonde factors; after specialisation, their product already yields the required square divisor. For prime $n=p$, we estimate the rank of the matrix modulo $p$. The required rank defect follows from a coefficient cancellation obtained from the involution $t\mapsto d\mathrm{/}t$ on ${\text{Fp}}^{\times }$ and the condition $\text{leg}dp=-1$.