A characterization of Sophie Germain primes

TL;DR

This paper characterizes Sophie Germain primes through the property of complete residue systems generated by a permutation of integers modulo n, establishing a new connection between prime properties and residue systems.

Contribution

It provides a novel characterization of Sophie Germain primes based on the permutation of integers forming complete residue systems modulo n.

Findings

Complete residue system characterization for odd n involving Sophie Germain primes

Partial results for even n cases

New link between prime properties and residue systems

Abstract

Let $n\ge 5$ be an odd integer. It is shown that $\{1^{\sigma(1)},\ldots,n^{\sigma(n)}\}$ is a complete residue system modulo $n$ for some permutation $\sigma$ of $\{1,\ldots,n\}$ if and only if $\frac{1}{2}(n-1)$ is a Sophie Germain prime. Partial results are obtained also for the case $n$ even.