Scott J. Beslin

Greatest common divisor matrices

Abstract Let S ={ x 1 , x 2 ,…, x n } be a set of distinct positive integers. Then n × n matrix [ S ]=( S ij ), where S ij =( x i , x j ), the greatest common divisor of x i and x j , is call the greatest common divisor (GCD) matrix on S. We initiate the study of GCD matrices in the direction of their structure, determinant, and arithmetic in Z n . Several open problems are posed.

A different type of differentiability

ABSTRACT The authors discuss the possibility of whether the tangent lines of a given function pass through specified points on the x-axis. Based on insightful student questions from calculus, they define and examine a differentiability property for functions possessing such tangent lines.

On Range and Reflecting Functions About the Line y = mx

Abstract The authors explore the importance of “range” and its relationship to continuously differentiable functions that have inverses when their graphs are reflected about lines other than y = x. Some open questions are posed for the reader.

Small Denominators: No Small Problem

It is always assumed that when either a or b is a fraction, it is in reduced form. The special case a= 9 and b = 7 appeared as a problem in [2]. As often happens in mathematics, the simplicity with which the problem is stated belies the complexity of solving it. We will observe interesting connections among the solution of (P), Farey sequences, and continued fractions. Many of these connections lead to good classroom problems in elementary number theory and computer science. Diligent readers will uncover some unanswered problems of their own. Is F(a, b) a function? The existence of a minimal denominator is ensured by the Well-Ordering Property of the natural numbers. We establish uniqueness of F(a, b) in the following proposition.

Primes and consecutive sums in arithmetic progressions

A necessary and sufficient condition for a positive integer to be prime is explored in terms of its number-theoretic ramifications and its generalization to arithmetic progressions. A computer experiment is shown describing how the main results were conjectured. Several open problems are posed for readers.