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.

This classroom note is presented as a suggested exercise--not to have the class prove or disprove Goldbach's Conjecture, but to stimulate student discussions in the classroom regarding proof, as well as necessary, sufficient, satisfied, and unsatisfied conditions. Goldbach's Conjecture is one of the oldest unsolved problems in the field of number…

The Pythagorean Theorem, arguably one of the best-known results in mathematics, states that a triangle is a right triangle if and only if the sum of the squares of the lengths of two of its sides equals the square of the length of its third side. Closely associated with the Pythagorean Theorem is the concept of Pythagorean triples. A "Pythagorean…

This paper takes a known point from Brocard geometry, a known result from the geometry of the equilateral triangle, and bring in Euler's [empty set] function. It then demonstrates how to obtain new Brocard Geometric number theory results from them. Furthermore, this paper aims to determine a [triangle]ABC whose Crelle-Brocard Point [omega]…

In some elementary courses, it is shown that square root of 2 is irrational. It is also shown that the roots like square root of 3, cube root of 2, etc., are irrational. Much less often, it is shown that the number "e," the base of the natural logarithm, is irrational, even though a proof is available that uses only elementary calculus. In this…

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…