Proof of three conjectures on determinants related to quadratic residues

ABSTRACT

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 ${\left|(i^2+d{j}^2)\left(\frac{i^2+d{j}^2}{n}\right)\right|}_{0\le i,j\le (n-1)/2},$ where d is any integer and $\left(\frac{\cdot }{n}\right)$ 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 $d\ne 0$ and n>2 are integers. Finally, for any odd prime p and integers c and d with $p\nmid cd$, we determine completely the Legendre symbol $\left(\frac{S_c(d,p)}{p}\right)$, where $S_c(d,p):={\left|\left(\frac{i^2+d{j}^2+c}{p}\right)\right|}_{1\le i,j\le (p-1)/2}$.

Keywords:

• Determinant
• divisibility
• Jacobi symbol
• Vandermonde-type determinant

Mathematics Subject Classifications:

• Primary 11C20
• Secondary 11A07
• 11A15
• 15A15

Acknowledgements

We thank Prof. Guo-Niu Han and the anonymous referee for helpful comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The second author is the corresponding author, and supported by the Natural Science Foundation of China [grant no. 11971222].