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 $n>3$ n greater than 3 and let c , d ∈ Z c,d∈ Z c comma d element of double struck upper Z . If n is composite, then det [ ( i 2 + c i j + d j 2 ) n − 2 ] 0 ≤ i , j ≤ n − 1 ≡ 0 ( mod n 2 ) align* [(i²+cij+dj²)ⁿ⁻²]0≤ i,j≤ n-1≡ 0 n² align* with no condition on c and d . If n = p $n=p$ n equals p is prime, the same congruence holds whenever the Legendre symbol ( d / p ) (d/p) left parenthesis d divided by p right parenthesis is