# On some determinants involving Jacobi symbols

## TL;DR

This paper investigates conjectures on determinants with Jacobi symbol entries, proving several cases for specific moduli and primes, using character sums over finite fields to establish the results.

## Contribution

The paper proves new conjectures by Sun regarding determinants with Jacobi symbol entries for specific moduli and primes, expanding understanding of their properties.

## Key findings

- Certain determinants are zero for specific moduli, confirming conjectures.
- Proved that $(10,9)_p=0$ for primes $p
ot ext{divisible by }12$.
- Established that $[5,5]_p=0$ for primes $p
ot ext{divisible by }20$.

## Abstract

In this paper we study some conjectures on determinants with Jacobi symbol entries posed by Z.-W. Sun. For any positive integer $n\equiv3\pmod4$, 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 $$(c,d)_n=\bigg|\left(\frac{i^2+cij+dj^2}n\right)\bigg|_{1\le i,j\le n-1}$$ and $$[c,d]_n=\bigg|\left(\frac{i^2+cij+dj^2}n\right)\bigg|_{0\le i,j\le n-1}$$ with $(\frac{\cdot}n)$ the Jacobi symbol. We also prove that $(10,9)_p=0$ for any prime $p\equiv5\pmod{12}$, and $[5,5]_p=0$ for any prime $p\equiv 13,17\pmod{20}$, which were also conjectured by Sun. Our proofs involve character sums over finite fields.