The Pi Mu Epsilon Journal
Abstracts and Authors - Spring 1999

The Primes Are a D-Complete Sequence

Paul S. Bruckman

Following Erdös and Lewin [1], we define a d-complete sequence A as an infinite increasing sequence of integers, such that no one element of A divides any other, and such that every sufficiently large integer is the sum of distinct elements of A. The purpose of this paper is to demonstrate that the sequence of primes is a d-complete sequence. In fact, a stronger result is actually demonstrated, namely that every integer m³ 12 is the sum of at least two distinct primes.

A New Insight Into the Goldbach Conjecture

Paul S. Bruckman

This paper is, in part, an expository paper encapsulating the more significant inroads that have been made toward an affirmative resolution of the famous Goldbach Conjecture (GC). Towards this end, the paper also presents a possible approach involving the square of the generating function , summed over all odd primes p. Finally, the paper postulates the existence of a minimal proper subset S of the odd primes p.

The Case of the Missing Case:

The Completion of a Proof by R. L. Graham

Julie Jones (Student) and Bruce F. Torrence

Randolf-Macon College

We present a corrected version to a proof of a theorem of Ronald Graham's from 1970 in the field of graph theory. The theorem gives a tight upper bound W(n) on the number of edges in a completely decomposable graph (for instance, any subgraph of the N-cube) with n vertices. The error in the proof is one of omission; a new argument is presented to cover the missing case, as well as an exposition of the remainder of the proof.

Divisibility Tests - Making Order Out of Chaos

Clayton W. Dodge

University of Maine

There are tests to check whether a given number is divisible by a given divisor, and it seems that each different divisor has its own test. For example, if the last digit of a base ten numeral is even, then the number is divisible by 2. If the sum of its digits is divisible by 3, then the number is divisible by 3. This paper shows that there is a theory that underlies this conglomeration of tests. All common divisibility tests, in fact, fall into two classes.

Determinants of Matrices Using the

Method of Generating Functions

Anthony Shaheen (Student)

Loyola Marymount University

A class of quasi-triangular matrices which are frequently encountered in stochastic systems is considered. The determinants of some special forms of matrices are found in terms of generating functions.

A Proof of the Pythagorean Theorem Using a Circle

Melissa Hicks and Beverly Collins (Students)

University of North Florida

A proof of the Pythagorean theorem is given by using a circle to establish an algebraic relation. The simplification of this algebraic relation is itself interesting.

Pythagorean Theorem

Tammy Muhs (Student)

University of North Florida

Two proofs of the Pythagorean theorem are given in this article. The first uses the area relation only. The second uses a combination of similar triangle and area relations.

The Cantor Shadow Problem:

Using Geometry to Compute Sums of Cantor Sets

Alan Koch and James Panariello

St. Edwards University and The University at Albany

Let C denote the Cantor set. Given any two real numbers a and b, we describe explicitly the set aC+bC. The description is obtained using a geometric argument, namely determining the shadow cast by the Cantor set of the x-axis when exposed to any light source emitting parallel rays. For nonzero a and b, the set aC+bC is the union of disjoint intervals, and it has the same basic structure as a partial Cantor set.

Factoring with a Braille 'n Speak

Wayne M. Dymàcek and Angie Matney (Student)

Washington and Lee University

In this paper, we explain what a Braille 'n Speak is, how it is used, and how it helped us factor expressions.

 Curious Numbers

 Xiaolong Ron Yu (Student)

Worcester Polytechnic Institute

An n-digit nonnegative integer x is called a curious number if and only if x2-x is divisible by 10n. The goal of this paper is to find explicit formulas for these numbers. By applying Euler's Theorem, two closely related modular arithmetic formulas are found. The formulas can be used to calculate all curious numbers for each n greater than one. Also the formulas show that there exist at most n-digit curious numbers for each n greater than one.