Proof of three conjectures on determinants related to quadratic residues

In this paper we confirm three conjectures of Z.-W. Sun on determinants. We first show that any odd integer n>3 divides the determinant (i2+dj2)i2+dj2n0≤i,j≤(n−1)/2, where d is any integer and (⋅n) is the Jacobi symbol. Then we prove some divisibility results concerning |(i+dj)n|0≤i,j≤n−1 and |(i2+dj2)n|0≤i,j≤n−1, where d≠0 and n>2 are integers. Finally, for any odd prime p and integers c and d with p∤cd, we determine completely the Legendre symbol Sc(d,p)p, where Sc(d,p):=i2+dj2+cp1≤i,j≤(p−1)/2.