MONTHLY, March 2006
Yeuh-Gin Gung and Dr. Charles Y. Hu Award to Hyman Bass for Distinguished Service to Mathematics
by Wayne Roberts
Did Euclid Need the Euclidean Algorithm to Prove Unique Factorization?
By David Pengelley and Fred Richman
Euclid proved that if a prime divides the product of two numbers, then it divides one of them. Or did he? We investigate what appears to be a serious gap in Euclid’s proof. In so doing, we run into some widespread misconceptions about the relationship between the celebrated Eudoxean theory of proportions of magnitudes and the presumably earlier theory of proportions of whole numbers.
On a Series of Goldbach and Euler
by Lluís Bibiloni, Jaume Paradís, and Pelegrí Viader
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Theorem 1 of Euler’s 1737 paper “Variae observationes circa series infinitas” states the astonishing result that the series of all unit fractions whose denominators are perfect powers of integers minus unity has sum one. Euler attributes the theorem to Goldbach. The proof is one of those examples of misuse of divergent series to obtain correct results so frequent during the seventeenth and eighteenth centuries. We examine this proof closely and, with the help of some insight provided by a modern (and completely different) proof of the Goldbach-Euler theorem, we present a rational reconstruction in terms that could be considered rigorous by modern Weierstrassian standards. At the same time, with the aid of a few ideas borrowed from nonstandard analysis, we see how the same reconstruction can be also be considered rigorous by modern Robinsonian standards. This last approach, though, is completely in tune with Goldbach and Euler’s proof. We thereby hope to convince the reader that a few simple ideas from nonstandard analysis serve to vindicate Euler’s work.
Infinitely Divisible Matrices
by Rajendra Bhatia
Many interesting matrices (like the Hilbert matrix, the Cauchy matrix, and the Pascal matrix) enjoy two kinds of positivity. Their entries are positive (a fact visible to the eye), and they are positive definite (a fact that needs proof, often by subtle arguments). Each of these matrices is endowed with a higher order of positivity. For every positive number r the matrix obtained by raising all entries to their rth power is also positive definite. This property is called infinite divisibility. The article discusses this special property in the context of these special matrices and in its relationship to other areas of mathematics —measures, Fourier transforms, the gamma function, and test matrices for numerical analysis.
Reflections on the Arbelos
by Harold P. Boas
Lost in the sands of time is the origin of the geometric figure named the arbelos by the ancient Greeks. The arbelos is the planar region bounded by three semicircles, tangent in pairs, with diameters lying on the same line. In this article, the author reflects on both the mathematics and the history of the arbelos.
Note on a Linear Difference Equation
by Robert E. Hartwig
Simple Norm Inequalities
by Lech Maligranda
The Pythagorean Theorem: What is it About?
by Alexander Givental
A Lattice-Ordered Skew Field is Totally Ordered if Squares are Postive
by YiChuan Yang
Problems and Solutions
A Tour Through Mathematical Logic
by Robert S. Wolf
Reviewed by Michael Beeson