A characterization of Sophie Germain primes - II

Abstract

Let $n\ge 4$ be an even integer. We show that $\{1^{\sigma (1)},2^{\sigma (2)},\dots ,n^{\sigma (n)}\}$ is a complete residue system modulo $n$ for some permutation $\sigma$ of $\{1,2,\dots ,n\}$ if and only if $p:=n/2$ is prime and $p-1$ is squarefree. This complements the results in [Int. J. Number Theory \textbf{14} (2018), 653--660], where an analogue characterization has been proved for odd integers.