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American Mathematical Monthly - November 2000


Even Order Regular Magic Squares are Singular
by R. Bruce Mattingly
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This work began as an attempt to characterize the eigenvalues and eigenvectors of magic squares. It turns out that a certain type of magic square may be deflated to form a skew-centrosymmetic matrix that has the same spectrum, except for one eigenvalue. Properties of Skew-centrosymmetric matrices are used to show that an even order regular magic square must have a zero eigenvalue. Alas, for odd order regular magic squares, this approach establishes only that the parity of a zero eigenvalue must be even.


The Factorial Function and Generalizations
by Manjul Bhargava
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The factorial function hardly needs any introduction. Starting with its fundamental interpretation as the number of ways n people can sit in n chairs, to its occurrence in formulae for binomial coefficients, Stirling numbers, and countless other combinatorial objects, it is indeed nearly impossible to study any area of combinatorics without becoming intimately familiar with the factorial. Perhaps it is due to this ubiquity in combinatorics that sometimes it is overlooked that the factorial also makes several important appearances in number theory! We take a closer look at some of these number-theoretic appearances, and thereby lead up to a series of generalizations of the factorial function, which recently have been applied to a variety of number-theoretic, ring-theoretic, and combinatorial problems.


Lost Theorems of Geometry
by Jason Jeffers
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The past century has seen geometry expand and diversify to the point where, today, very few areas of mathematics are not touched in some way by concepts that have their origin in classical geometry. We discuss, and prove, an attractive class of theorems that were left behind by this whole process. Rediscovering them is akin to chancing upon sunken treasure. As an example of one such result: a bijection of the hyperbolic plane that preserves hyperbolic straight lines must necessarily be an isometry.


Sums of Squares of Polynomials
by Walter Rudin
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This is an exposition of one of Hilbert's early papers. The question was: If P is a positive polynomial, is (even) degree d, in n real variables, must P then be a sum of squares of polynomials? The answer is: Yes when n = 1, yes when d = 2, yes when n = 2, d = 4 (this is the hard case), but no for all other pairs (n, d).


Diophantine Olympics and World Champions: Polynomials and Primes Down Under
by Edward B. Burger
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The name of the game is finding the world champion sequence of best approximations for an irrational number. But which sequences have what it takes to be world champions? Can sequences get in shape and shed unwanted terms to win the gold? Here these questions are addressed together with some basic diophantine approximation and some new open questions. Let the games begin.



A Four Vertex Theorem for Polygons
by Serge Tabachnikov
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A Note on the Metrical Theory of Continued Fractions
by Glyn Harman and Kam C. Wong
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Multidimensional Analytic Deflation
by Peter Kravanja and Ann Haegemans

Irrationality of the Square Root of Two - A Geometric Proof
by Tom M. Apostol
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Fundamental Theorem of Algebra - Yet another Proof
by Anindya Sen
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Bolzano, Cauchy, Epsilon, Delta

by Walter Felscher



The Wild Numbers
By Philibert Schogt

Uncle Petros and Goldbach's Conjecture
By Apostolos Doxiadis

Reviewed by Steve Kennedy
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Proofs from THE BOOK
By Martin Aigner and Gunter M. Ziegler

Reviewed by Marion D. Cohen