Rotation Symmetries of Sequential Matrices with Applications to the Jacobi Symbol

TL;DR

This paper explores symmetry properties of sequential matrices over certain modular rings and extends these symmetries to the Jacobi symbol, revealing new algebraic relationships regardless of primality.

Contribution

It identifies and generalizes symmetry properties of matrices and Jacobi symbols over rings where the modulus is of the form n^2+1, independent of primality.

Findings

Symmetries of sequential matrices over Z/(n^2+1)Z are characterized.

Generalized symmetry relations involving the Jacobi symbol are established.

Results hold regardless of whether n^2+1 is prime.

Abstract

Suppose that $p$ is an odd prime and $\genfrac{(}{)}{}{}{\cdot}{p}$ denotes the Legendre symbol modulo $p$. If $p$ is has the form $p= n^2+1$ then one easily verifies that $\genfrac{(}{)}{}{}{a}{p} = \genfrac{(}{)}{}{}{-a}{p}$ 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.