This article presents a method of reducing fractions without factoring. The ideas presented may be useful as a project for motivated students in an undergraduate number theory course. The discussion is related to the Euclidean Algorithm and its variations may lead to projects or early examples involving efficiency of an algorithm.

Mathematical historians place Heron in the first century. Right-angled triangles with integer sides and area had been determined before Heron, but he discovered such a "non" right-angled triangle, viz 13, 14, 15; 84. In view of this, triangles with integer sides and area are named "Heron triangles." The Indian mathematician Brahmagupta, born in…

The number Pi has a rich and colorful history. The origin of Pi dates back to when Greek mathematicians realized that the ratio of the circumference to the diameter is the same for all circles. One is most familiar with many of its applications to geometry, analysis, probability, and number theory. This paper demonstrates several examples of how…

The sequence 1, 1, 2, 3, 5, 8, 13, 21, ..., known as Fibonacci sequence, has a long history and special importance in mathematics. This sequence came about as a solution to the famous rabbits' problem posed by Fibonacci in his landmark book, "Liber abaci" (1202). If the "n"th term of Fibonacci sequence is denoted by [f][subscript n], then it may…

In this paper, the author gives a further simple generalization of a power series evaluation of an integral using Taylor series to derive the result. The author encourages readers to consider numerical methods to evaluate the integrals and sums. Such methods are suitable for use in courses in advanced calculus and numerical analysis.

Focuses on the study of the sum of two integer squares, neither of which is zero square. Develops some new interesting and nonstandard ideas that can be put to use in number theory class, mathematics club meetings, or popular lectures. (ASK)

A triple (x,y,z) of natural numbers is called a Primitive Pythagorean Triple (PPT) if it satisfies two conditions: (1) x[squared] + y[squared] = z[squared]; and (2) x, y, and z have no common factor other than one. All the PPT's are given by the parametric equations: (1) x = m[squared] - n[squared]; (2) y = 2mn; and (3) z = m[squared] +…

Pascal's Triangle is, without question, the most well-known triangular array of numbers in all of mathematics. A well-known algorithm for constructing Pascal's Triangle is based on the following two observations. The outer edges of the triangle consist of all 1's. Each number not lying on the outer edges is the sum of the two numbers above it in…

We present an expository account of the development of the theory of binary quadratic forms. Beginning with the formulation and proof of the Two-Square Theorem, we show how the study of forms of the type x[squared] + ny[squared] led to the discovery of the Quadratic Reciprocity Law, and how this theorem, along with the concept of reduction relates…

Discusses the mathematical properties of magic squares. Includes a computer program that will test magic square criteria. (PK)

Identifies and proves two theorems related to expressing rational numbers in decimal form for any natural base m>1. Includes two BASIC computer programs with sample runs. (MVL)