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American Mathematical Monthly - June/July 2001



The Theorem of Pappus: A Bridge between Algebra and Geometry
by Elena Anne Marchisotto

A theorem of the great mathematician, Pappus of Alexandria, makes a beautiful connection between algebra and geometry that we explore in this article. We start with a geometric structure and impose certain postulates and theorems to determine an algebraic structure of an abstract coordinate set. Then we prove that Pappus' theorem is sufficient for commutativity of multiplication there. It can also be proved that it is necessary. At the end of the article we examine a host of horizons opened by Pappus' theorem and provide a substantial resouce list for further exploration.


A Curious Connection Between Fermat Numbers and Finite Groups
by Carrie E. Finch and Lenny Jones,

The authors were investigating finite groups possessing a property related to certain subsets of the group, when suddenly the problem became entangled in number theory. The shocking conclusion is that the solution to a large part of the group theory problem is a direct consequence of the fact that the Fermat number 225 +1 is not prime.


Subtangents—an Aid to Visual Calculus
by Tom M. Apostol and Mamikon A. Mnatsakanian,

Many standard problems in calculus can be easily solved by an innovative visual approach that makes no use of formulas. The method, developed by Mamikon Mnatsakanian, relies on geometric intuition and is easily understood by very young students. It can be used to find (without calculus) areas of many plane regions, including: an oval ring, a parabolic segment, the region under an exponential curve, the region under one arch of a cycloid, the region under a tractrix, and the region between two curves traced out by the rear and front wheels of a bicycle. The treatment of the parabolic segment and the exponential (which appear elsewhere) use geometric properties of subtangents to these curves. This paper explores properties of subtangents in greater detail. It shows how subtangents can be used in practice to draw tangent lines, and it reveals an unexpected connection between pursuit curves and exponential curves. The paper also contains a rigorous justification of Mamikon's geometric method.


An Elementary Problem Equivalent to the Riemann Hypothesis
by Jeffrey C. Lagarias

The Riemann hypothesis is shown to be equivalent to the following statement: Let

Hn=Σ n 1/j

be the n-th harmonic number. Show, for each n≥1, that

ΣdHn+exp(Hn) log(Hn),

with equality only for n=1. This equivalence is deduced from a criterion of G. Robin for the Riemann hypothesis. The paper describes the history of earlier work leading to Robin's criterion, including that of Ramanujan on highly composite numbers, and Alaoglu and Erdös on colossally abundant numbers.


Two Classical Surprises Concerning the Axiom of Choice and the Continuum Hypothesis
by Leonard Gillman

The two surprises are that Trichotomy (defined in a moment) implies the Axiom of Choice and the General Continuum Hypothesis implies the Axiom of Choice. (Trichotomy is the proposition that of any two cardinals, a and b, either a<b or a=b or a>b.) The result for Trichotomy was proved by F. Hartogs in 1915 and the result for the General Continuum Hypothesis was established by W. Sierpinski in 1947. The proofs presented in the paper are based on the original proofs and are intended for the nonspecialists.


Problems and Solutions


A Modified Problem of Pillai and Some Related Questions
by G.E. Hardy and M.V. Subbarao,

Some Curious Sequences Involving Floor and Ceiling Functions
by M.A. Nyblom

The Right-hand Derivative of an Integral
by Timothy G. Feeman and Osvaldo Marrero,


Introduction to Topological Manifolds
by John M. Lee
Reviewed by Claude LeBrun

Trigonometric Delights
by Eli Maor
Reviewed by George Swift

The Universal Computer: the Road from Leibniz to Turing
by Martin Davis

The Universal History of Computing: From the Abacus to the Quantum Computer
by Georges Ifrah

Reviewed by Gerald B. Folland