Abstracts and Authors - Spring 1999

**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 x^{2}-x is divisible by 10^{n}. 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.