#### Vol. 8, No. 4, 2015

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

### Karenna Genzlinger and Keir Lockridge

Vol. 8 (2015), No. 4, 653–663
##### Abstract

In the paper “What is special about the divisors of 24?”, Sunil Chebolu proved an interesting result about the multiplication tables of ${ℤ}_{n}$ from several different number theoretic points of view: all of the 1s in the multiplication table for ${ℤ}_{n}$ 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 ${ℤ}_{n}$ if and only if $p$ is a Sophie Germain prime.

##### Keywords
Sophie Germain primes, group of units, Gauss–Wantzel theorem
Primary: 11A41
Secondary: 16U60
##### Milestones
Received: 9 June 2014
Revised: 9 June 2014
Accepted: 15 July 2014
Published: 23 June 2015

Communicated by Kenneth S. Berenhaut
##### Authors
 Karenna Genzlinger Department of Mathematics Gettysburg College Gettysburg, PA 17325 United States Keir Lockridge Department of Mathematics Gettysburg College Gettysburg, PA 17325 United States