Volume 41 - Issue 1 - June 2016


Major milestones in the twin prime conjecture

Anthony Breitzman, Sr.


In April 2013 Yitang Zhang announced a proof that there are infinitely many pairs of prime numbers that have a difference of at most 70 million (see Zhang (2014)). Others have since narrowed the gap from 70 million to just 246. Reducing the gap to 2 would prove the twin prime conjecture. The significance is a path to solving an ancient problem that looked hopeless just a few years ago. The popular press has discussed Zhang's result but at a very high level. Zhang (2014) and subsequent papers appearing in number theory journals present only the most recent details and are not really accessible to the nonspecialist. This paper attempts to bridge the gap and present a history of major milestones leading to the current state of the twin prime conjecture written at the level of working mathematicians.

Applying a Möbius transformation for solving quartic equations

Raghavendra G. Kulkarni


We propose a new method for solving a quartic equation, wherein the given quartic equation is transformed into a reciprocal equation using a Möbius transformation. We then use the property of reciprocal equations to solve the quartic equation.

Probabilistic model of cryptographic key collisions in relation to the key length

T. Van Hecke


Cryptographic keys are used to encipher data. Agencies generating these key numbers want to combine a minimal key length for computational and data storage reasons with a guaranteed service of unique key numbers. This paper describes the modeling of the length of the key number to reach an imposed collision improbability.

Lunar rhythms and strange signatures

Andrew Simoson


We show that the phases of the Moon loosely cycle with short period 19 years and long period 141 years using a simple model referred to as a strange signature, the analysis of which is equivalent to a continued fraction algorithm. Counterintuitively, the variation in the shorter period appears to be over five times larger than the variation in the longer period. Furthermore, we explore how these periods change as the Moon slips away from the Earth.

A sequence of good approximations for the period of a pendulum with large initial amplitude

Thomas J. Osler and Jesse M. Kosior


We present three elementary approximate formulas for the period of a pendulum which starts at rest from a large angle of displacement. The first of these formulas is known, but the other two may be new. These three formulas result from taking the first three partial products of a new infinite product of nested radicals for the complete elliptic integral of the first kind that gives the exact period. Thus, more elementary approximations can be obtained from this exact product, but they become increasingly complex. Therefore, we stopped at three. We give a detailed table clearly displaying the accuracy of the approximations over the full range of possible initial angles of displacement. This infinite product of nested radicals is a special case of a new infinite product for the arithmetic–geometric mean that has appeared recently.

Constructing integer matrices with integer eigenvalues

Christopher Towse and Eric Campbell


In spite of the provable rarity of integer matrices with integer eigenvalues, they are commonly used as examples in introductory courses. We present a quick method for constructing such matrices starting with a given set of eigenvectors. The main feature of the method is an added level of flexibility in the choice of allowable eigenvalues. The method is also applicable to nondiagonalizable matrices, when given a basis of generalized eigenvectors. We have produced an online web tool that implements these constructions.

Perimeters of Fermat ovals

Khaldoun El Khaldi and Elias G. Saleeby


A basic question in integral geometry is how to compute the perimeter of a set in the Euclidean plane. In this article, we examine the problem of computing the perimeters for the Fermat family of curves of even degree. We first employ a classical approach and derive a series representation for the perimeters. Then we examine the use of the Cauchy intergral formula for the perimeter of a closed convex set. This gives us an approximate simple formula in closed form, and suggests a novel elmentary Monte Carlo method to estimate the perimeters. Finally we employ a quasi Monte Carlo method based on the Cauchy–Crofton formula to compute the perimeters.

The friendship paradox

Yang Cao and Sheldon M. Ross


We clarify and generalize the friendship paradox. Among other things we show that a randomly chosen friend of a randomly chosen person X has stochastically more friends than X does.

Factorial, raw and central moments

S. Nadarajah and I. E. Okorie


General relations expressing factorial moments in terms of raw moments, raw moments in terms of factorial moments, factorial moments in terms of central moments, and central moments in terms of factorial moments are derived. Examples are given.