#### Roles

Karenna Genzlinger: Class of 2014

#### Document Type

Article

#### Publication Date

2015

#### Department

Mathematics

#### Abstract

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

#### Copyright Note

This is the publisher's version of the work. This publication appears in Gettysburg College's institutional repository by permission of the copyright owner for personal use, not for redistribution.

#### Recommended Citation

Genzlinger, Karenna and Keir Lockridge. "Sophie Germain Primes and Involutions of Znx." Involve 8.4 (2015), 653-663.

#### Required Publisher's Statement

Original version is available from the publisher at: http://msp.org/involve/2015/8-4/p08.xhtml