On some determinants involving Jacobi symbols

Abstract

In this paper we study some conjectures on determinants with Jacobi symbol entries posed by Z.-W. Sun. For any positive integer $n\equiv 3(\mathrm{mod}4)$, 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 $$(c,d{)}_n=|\left(\frac{i^2+cij+d{j}^2}{n}\right){|}_{1\le i,j\le n-1}$$ and $$[c,d{]}_n=|\left(\frac{i^2+cij+d{j}^2}{n}\right){|}_{0\le i,j\le n-1}$$ with $(\frac{\cdot }{n})$ the Jacobi symbol. We also prove that $(10,9{)}_p=0$ for any prime $p\equiv 5(\mathrm{mod}12)$, and $[5,5{]}_p=0$ for any prime $p\equiv 13,17(\mathrm{mod}20)$, which were also conjectured by Sun. Our proofs involve character sums over finite fields.

22 pages