Sophie Germain primes and involutions of ℤn×

Genzlinger, Karenna; Lockridge, Keir

doi:10.2140/involve.2015.8.653

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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

DOI: 10.2140/involve.2015.8.653

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

Mathematical Subject Classification 2010

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

Vol. 8, No. 4, 2015