Some determinants involving binary forms

TL;DR

This paper investigates the divisibility properties of determinants formed from Jacobi symbols of quadratic forms, confirming a conjecture about their divisibility by the square of Euler's totient function for certain odd integers.

Contribution

It proves a conjecture relating determinants of quadratic form-based Jacobi symbols to divisibility by (n)^2 for specific odd integers, advancing understanding of these arithmetic properties.

Findings

Determinants involving Jacobi symbols of quadratic forms are divisible by (n)^2 under certain conditions.

Confirmed a conjecture about the divisibility of these determinants for odd integers with specific Legendre symbol properties.

Provides new insights into the structure of determinants formed from quadratic forms and their arithmetic properties.

Abstract

In this paper, we study arithmetic properties of certain determinants involving powers of $i^2+cij+dj^2$, where $c$ and $d$ are integers. For example, for any odd integer $n>1$ with $(\frac dn)=-1$ we prove that $\det [ (\frac{i^2+cij+dj^2}{n})]_{0\le i,j\le n-1}$ is divisible by $\varphi(n)^2$, where $(\frac{\cdot}{n})$ is the Jacobi symbol and $\varphi$ is Euler's totient function. This confirms a previous conjecture of the second author.