Mathematics Magazine - February 2005

ARTICLES

Fermat: The Founder of Modern Number Theory
Israel Kleiner
3-14
What were some of Fermat’s major number-theoretic results? What inspired his labors? Why did he not publish his proofs? Did he have a proof of Fermat’s Last Theorem? How should we view his results in number theory in the light of the work of subsequent centuries? These are among the questions addressed in this paper.

PROOF WITHOUT WORDS

José Gomez
Pythagorean Triples and Factorization of Even Squares
14
A diagram shows a one-to-one correspondence between Pythagorean triples and factorizations of even squares of the form p² = 2 nm.

The Singled Out Game
Kennan Shelton
15-25
The music television cable channel MTV is not normally associated with the inspiration of intellectual activity. However, its dating game show “Singled Out” does suggest a mathematically interesting game. In the Singled Out game, players attempt to guess the outcomes of flips of a coin, but announce their guesses in a fixed order and so that all can hear. Thus later players hear previous guesses and have extra information at the time of their guess. Can the players use this information to their advantage? In this article, we use Catalan numbers to analyze the two-player version of the Singled Out game and provide some experimental results for the three-player version.

Height and Excess of Pythagorean triples
Darryl McCullough
26-44
The set of Pythagorean triples has a lot of interesting structure, which has intrigued both amateur and professional mathematicians. It is the topic of an extensive mathematical literature, almost all of which relies on an enumeration of primitive Pythagorean triples that has been known since ancient times. But it is not widely known that there is a different enumeration, based on two simple geometric parameters that we call the height and the excess. In this article, we use these parameters to make some known results about Pythagorean triples more transparent. And we use them to achieve a better understanding of one natural group structure on the set of primitive Pythagorean triples, and to discover another one.

NOTES

Hidden Group Structure
Ruth I. Berger
45-48
If you have ever wondered how a set like {5, 15, 25, 35} can possibly turn out to be a group under multiplication modulo 40, this paper will be of interest to you. It examines the underlying group theory that makes this possible, and explains how to easily produce your own examples of this type. Knowing about these groups with no obvious identity element provides a great resource to any Abstract Algebra instructor or the curious student.

The St. Basil’s cake problem
Christina Savvidou
48-51
Perhaps the most popular cake in the Greek world during the Christmas period is not the well-known Christmas cake, but rather a cake called the St. Basil’s cake. The cake is prepared using simple ingredients like flour, eggs, orange juice, etc, but also contains a coin wrapped in aluminum foil, that nobody knows where it is. With the arrival of the New Year, the cake is cut (with a knife) into sectors of the circle. Each member of the family takes a sector and the one who finds the coin, according to the tradition, is considered to be luckiest one of the New Year. However, sometimes, while the cake is cut into sectors, the knife hits the coin. This event happens quite often, and as the note shows the probability of such an event is quite high. This problem can be considered as an extension of the Buffon’s needle problem.

Replacement Costs: The Inefficiencies of Sampling with Replacement
Emily S. Murphree
51-57
When a sample is chosen with replacement, the same item may be chosen several times. The number of distinct items appearing in the sample is a random variable whose properties are explored in this paper. To avoid combinatorial difficulties, the distribution of the number of distinct items is derived by using a Markov chain approach. Expected values are derived from simple properties of Bernoulli random variables. Finally, the distributional results are used to test a random number generator.

 

Can the Committee Meet? A Markov Chain Analysis
Terry L. Kiser, Thomas A. McCready, and Neil C. Schwertman
57-63
Scheduling committee meetings when all members are available can be quite challenging. In this paper we use probability and a Markov chain approach to determine the likelihood of all members being available simultaneously. Furthermore, this analysis shows, in agreement with intuition, that as the number of committee members and their teaching loads increase, it becomes much more difficult to find an available time for a meeting.

Dirichlet and Fresnel Integrals via Iterated Integration
Paul Loya
63-67
Two important improper integrals that seem to pop up more than usual are the celebrated Dirichlet and Fresnel integrals. In this paper, we prove a theorem that gives conditions under which one can iterate improper integrals and then we use this theorem to derive the Dirichlet and Fresnel integrals. This theorem can also be used to derive "generalized" Fresnel integrals and prove the renowned Weierstrass M-Test.