 # American Mathematical Monthly -March 2008 ## March 2008

Primitive Juggling Sequences
By: Fan Chung and Ron Graham
fan@ucsd.edu, graham@ucsd.edu
For the past several decades, mathematicians (and mathematically inclined jugglers) have investigated a new way of interpreting certain finite integer sequences as possible patterns which in theory could be juggled. In particular, one can define a directed graph on these sequences which describes how one can transition from one pattern to another one by throwing one more ball. In this note, we show how to enumerate the number of cycles of each length in this graph, starting at any particular vertex. We also treat the case in which the cycle is required to be primitive, by which we mean that it does not return to the starting vertex until the last step. Universal Measuring Boxes with Triangular Bases
By: Jin Akiyama, Hiroshi Fukuda, Chie Nara, Toshinori Sakai, and Jorge Urrutia
fwjb5117@mb.infoweb.ne.jp, fukuda@u-shizuoka-ken-ac.jp, cnara@ried.tokai.ac.jp, tsakai@ried.tokai.ac.jp, urrutia@matem.unam.mx
Measuring cups are everyday instruments used to measure the amount of liquid required for many common household tasks such as cooking. A measuring cup usually has gradations marked on its sides. In this paper we study measuring boxes without gradations which can nevertheless measure any integral amount, say liters, of liquid up to their full capacity. These boxes will be called universal measuring boxes. We study two types of measuring boxes with triangular bases and plane quadrilateral sides, and for each type determine its maximum possible capacity. Qualitative Tools for Studying Periodic Solutions and Bifurcations as Applied to the Periodically Harvested Logistic Equation
By: Diego M. Benardete, V. W. Noonburg, and B. Pollina
dbenarde@cs.hartford.edu, noonburg@hartford.edu, pollina@hartford.edu
One-dimensional differential equations that are periodic in time arise in many areas of applied mathematics. The attracting and repelling periodic solutions and their bifurcations are the key qualitative features that determine the observed behavior of the system being modeled. Simulations to study these systems are guided by qualitative theoretical results and techniques. Several of these tools are presented using the familiar and intuitive logistic population model with periodic harvesting. For one such equation, we use these tools to answer a question posed by Campbell and Kaplan about the bifurcation value and then employ symmetry techniques to rigorously validate an efficient algorithm for its computation. We also discuss the inspiring yet wavering history of science and mathematics in three places: the development of the logistic equation and its variants, the use of Poincaré's first return map in dynamics and chaos theory, and the progress and lack of it that marks the investigation of Hilbert's still unsolved sixteenth problem. On a4+b4+c4+d4 = (a+-b+c+d)4
By: Lee W. Jacobi and Daniel J. Madden
We show by explicit construction that Euler's Diophantine equation a4+b4+c4+d4=e4 has infinitely many integral solutions with all terms non-zero. While the arithmetic geometry of elliptic curves provides a context and mathematical description of the methods used, these methods involve mostly elementary algebra. All calculations are carried out explicitly, and any reader could easily use them to find additional new solutions.

Euler's equation is specialized to a4+b4+c4+c4= (a+b+c+d)4 for which one solution is already known. This equation is rewritten into a pair of equations that describe a family of elliptic curves, and we show that the known solution is a non-torsion point on its member of the family. That is to say that the known solution is a point of infinite order in the Abelian group formed on its particular elliptic curve. This tells us that the equation has infinitely many solutions.

While this construction can be used to find new Euler solutions, those it produces are very large. To find solutions of more moderate size, we add two additional variables. These allow us to construct an equation in 6 homogeneous variables that describes a higher dimensional surface covering the surface given by the specialized Euler equation. We give an explicit formula for transforming solutions to the new equation into solutions of Euler's equation. The new equation is such that any one solution easily leads to many others. The construction provides any number of ways of changing one solution into another. Thus it is possible to add more examples to previously known 88 solutions to Euler's equation. The paper ends with a list of new solutions to Euler's equation obtained this way. Reciprocity Relations for Bernoulli Numbers
By: Takashi Agoh and Karl Dilcher
agoh_takashi@ma.noda.tus.ac.jp, dilcher@mathstat.dal.ca
This note takes a fresh look at a certain class of identities satisfied by Bernoulli numbers. Its purpose is threefold: First we reinterpret and rewrite some known identities as reciprocity relations for Bernoulli numbers. Second, we derive an apparently new identity which can be interpreted as a three-term reciprocity relation. As a consequence we obtain a surprising new quadratic recurrence relation, and we indicate how more such relations can be obtained. Our third purpose is to point the reader to the very rich classical literature on Bernoulli numbers. In fact, it turns out that almost all results quoted in this note were known in the 19th century, and were often rediscovered several times. Notes

The Fubini Principle
By: Krassimir Penev
krassi@att.net

A Refinement of Raabe's Test
By: Franciszek Prus-Wisniowski
wisniows@univ.szczecin.pl

Arithmetic Progressions and Binary Quadratic Forms
By: Ayse Alaca, Saban Alaca, and Kenneth S. Williams
aalaca@math.carleton.ca, salaca@math.carleton.ca, williams@math.carleton.ca

On Sums of Consecutive Integral Roots
By: Peter W. Saltzman and Pingzhi Yuan
pwsaltzman@leonardcarder.com, mcsypz@mail.sysu.edu.cn

Reviews

Inexhaustibility: A Non-Exhaustive Treatment and Gödel's Theorem: An Incomplete Guide to Its Use and Misuse
By: Torkel Franzén
Reviewed by: Martin Davis
martin@eipye.com

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