On some determinants involving Jacobi symbols

In this paper we study some conjectures on determinants with Jacobi symbol entries posed by Z.-W. Sun. For any positive integer n≡3(mod4), we show that(6,1)n=[6,1]n=(3,2)n=[3,2]n=0 and(4,2)n=(8,8)n=(3,3)n=(21,112)n=0 as conjectured by Sun, where(c,d)n=|(i2+cij+dj2n)|1≤i,j≤n−1 and[c,d]n=|(i2+cij+dj2n)|0≤i,j≤n−1 with (⋅n) the Jacobi symbol. We also prove that (10,9)p=0 for any prime p≡5(mod12), and [5,5]p=0 for any prime p≡13,17(mod20), which were also conjectured by Sun. Our proofs involve character sums over finite fields.