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

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

Sophie Germain primes, group of units, Gauss–Wantzel theorem
Mathematical Subject Classification 2010
Primary: 11A41
Secondary: 16U60
Received: 9 June 2014
Revised: 9 June 2014
Accepted: 15 July 2014
Published: 23 June 2015

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