Curious consequences of a Miscopied Quadratic
Jeffrey L. Poet and Donald L. Vestal, Jr.
The starting point of this article is a search for pairs of quadratic polynomials x2 + bx ± c with the property that they both factor over the integers. The search leads quickly to some number theory in the form of primitive Pythagorean triples, and this paper develops the connection between these two topics.
The Platonic Solids from their Rotation Groups
The five Platonic solids are constructed (as graphs) from their rotational symmetry groups. The constructions are based on an idea of Bertram Kostant and are quite simple; conjugacy classes in the group are the vertices of the graphs and products determine adjacency.
On Primes, Density Measures, and Statistical Independence
A result known as the Borel-Cantelli lemma is about probabilities of sequences of events. This article presents an example in which it appears that the hypotheses of the lemma are satisfied but the conclusion is not. The explanation of why not combines elements of probability theory, number theory, and analysis.
Mark Bridger and Andrei Zelevinsky
Within the set of points in the plane with integer coordinates, one point is said to be visible from another if no other point in the set lies between them. This study of visibility draws in topics from a wide variety of mathematical areas, including geometry, number theory, probability, and combinatorics.
"Mathematical Games" and Beyond: Part II of an Interview with Martin Gardner
This is a continuation of an interview (begun in the May issue) with Martin Gardner, in which he discusses his “Mathematical Games” columns in Scientific American and some of his other writings. He goes on to share his views on a variety of aspects of science and mathematics.
Jeffrey A. Graham
The polynomials studied here have the property that . The main results concern their density on a given interval.
A Variant of the Patition Function
John F. Loase, David Lansing, Carrie Hryczaniuk, and Jamie Cahoon
This note concerns the number of ways that a positive integer can be expressed as a sum of primes. Some elementary results on this variant of the partition function and some questions for further study are raised.
Exactly When is (a + b)n ≡ an + bn(modn)?
Pratibha Ghatage and Brian Scott
Although primes satisfy the congruence in the title, they are not the only numbers that do. It is shown that a necessary and sufficient condition is that Fermat’s little theorem hold for n.
How Do You Slice the Bread?
James Colin Hill, Gail Nord, Eric Malm, and John Niord
The bread in question is like a standard loaf with a rounded top, and the cut to be made is to be diagonal, thereby giving two pieces that are roughly triangular in shape. Two problems are discussed: how to make the cut so that the two parts have an equal amount of bread, and how to do it so the amounts of crust are the same.
Limits of Functions of Two Variables
An example is given of a function that is discontinuous at the origin, but for which the limit along every curve of the form y = xn is 0. The function is a modification of the standard examples of this type given in textbooks, but has this much stronger path property.