A characterization of Sophie Germain primes

Abstract

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