Rotation Symmetries of Sequential Matrices with Applications to the Jacobi Symbol

Abstract

Suppose that $p$ is an odd prime and $\left(\frac{\cdot }{p}\right)$ denotes the Legendre symbol modulo $p$. If $p$ is has the form $p=n^2+1$ then one easily verifies that $\left(\frac{a}{p}\right)=\left(\frac{-a}{p}\right)$ for all $a\in \mathbb{Z}/p\mathbb{Z}$. We identify various symmetry properties of sequential matrices over $\mathbb{Z}/(n^2+1)\mathbb{Z}$ regardless of whether $n^2+1$ is prime. We deduce from these results a collection of symmetries involving Jacobi symbol modulo $n^2+1$ which generalize our above observation on the Legendre symbol.