Sophie Germain primes and involutions of $\mathbb{Z}_n^\times$

Abstract

In the paper “What is special about the divisors of 24?”, Sunil Chebolu proved an interesting result about the multiplication tables of ${\mathbb{Z}}_n$ from several different number theoretic points of view: all of the 1s in the multiplication table for ${\mathbb{Z}}_n$ are located on the main diagonal if and only if $n$ is a divisor of 24. Put another way, this theorem characterizes the positive integers $n$ with the property that the proportion of 1s on the diagonal is precisely 1. The present work is concerned with finding the positive integers $n$ for which there is a given fixed proportion of 1s on the diagonal. For example, when $p$ is prime, we prove that there exists a positive integer $n$ such that $1∕p$ of the 1s lie on the diagonal of the multiplication table for ${\mathbb{Z}}_n$ if and only if $p$ is a Sophie Germain prime.