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American Mathematical Monthly - March 2017

The March issue of the Monthly comes in like a (friendly) lion: Pete Clark uses Euclid’s proof of the infinitude of primes to develop a general algebraic criterion for an integral domain to have infinitely many nonassociate irreducible elements; Joseph Bak and Strahmir Popovassilev offer a survey of Cauchy’s closed curve theorem from A. Clairaut (prehistory) to D. J. Newman (modern times); Jonathan Sondow and Kieran MacMillan define primary pseudoperfect numbers and discuss connections to arithmetic progressions and solutions to the Erdős–Moser Diophantine equation; and Koji Momihara and Masashi Shinohara study finite sets of points on the unit circle having a fixed number of distinct distances between pairs of points.

In the Notes section, Michael Weiss locates the rational squares, Shubhodip Mondal gives conditions for polynomials and entire functions to induce surjections on matrix algebras, Dirk Huylebrouck provides a striking example of a new infinite, regular, compound polyhedron, and Joseph Plante gives an historically-minded proof of the mean value theorem (à la Cauchy).

There are, as ever, problems to solve. Dominic Lanphier reviews Avner Ash’s and Robert Gross’s Summing It Up: From One Plus One to Modern Number Theory, the third in their series of books introducing sophisticated and important arithmetic tools to a broad audience.

  — Susan Jane Colley, Editor


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Table of Contents

Yueh-Gin Gung and Dr. Charles Y. Hu Award for 2017 to Martha Siegel for Distinguished Service to Mathematics

p. 195.

Annalisa Crannell and Doug Ensley


The Euclidean Criterion for Irreducibles

p. 198.

Pete L. Clark

We recast Euclid’s proof of the infinitude of prime numbers as a Euclidean criterion for a domain to have infinitely many atoms. We make connections with Furstenberg’s “topological” proof of the infinitude of prime numbers and show that our criterion applies even in certain domains in which not all nonzero nonunits factor into products of irreducibles.


The Evolution of Cauchy's Closed Curve Theorem and Newman's Simple Proof

p. 217.

Joseph Bak and Strashimir G. Popvassilev

We examine the development of Cauchy’s closed curve theorem, including the early contributions of Clairaut, d’Alembert, Cauchy himself, Goursat, and Pringsheim, as well as more recent approaches due to Ahlfors, Rudin, and others. A particularly simple proof was given by D. J. Newman, utilizing his original definition of a simply-connected region in the (complex) plane. We show that this definition is equivalent to the other, more familiar definitions of simple-connectedness so that Newman’s approach offers an alternative and very elegant proof of the general result.


Primary Pseudoperfect Numbers, Arithmetic Progressions, and the Erdős-Moser Equation

p. 232.

Jonathan Sondow and Kieren MacMillan

A primary pseudoperfect number (PPN) is an integer K > 1 such that the reciprocals of K and its prime factors sum to 1. PPNs arise in studying perfectly weighted graphs and singularities of algebraic surfaces, and are related to Sylvester’s sequence, Giuga numbers, Zn´am’s problem, and Curtiss’s bound on solutions of a unit fraction equation. 
In this paper, we show that K is congruent to 6 modulo 36 if 6 divides K, and uncover a remarkable 7-term arithmetic progression of residues modulo 288 in the sequence of known PPNs. On that basis, we pose a conjecture which leads to a conditional proof of a new record lower bound on any nontrivial solution to the Erdős-Moser Diophantine equation.


Distance Sets on Circles

p. 241.

Koji Momihara and Masashi Shinohara

An n-point k-distance set on the unit sphere St ⊂ ℝt+1 is a set X of n points on St such that exactly k Euclidean distances occur between two distinct points in X. In this paper we treat distance sets on S1, and show that if k is sufficiently small relative to n, then X lies on a regular polygon. More precisely, we prove that for an n-point k-distance set X on S1 with n ≥ 4, if k < 3t or 3t − 2 according to whether n = 4t, 4t − 1 or n = 4t − 2, 4t − 3, respectively, then X lies on a regular 2k or (2k + 1)-sided polygon. Furthermore, we see that this bound cannot be further improved. In addition, we find an application of Kneser’s addition theorem to distance sets on circles.



Where Are the Rational Squares?

p. 255.

Michael Weiss

We study the minimal denominator among all rational numbers whose square lies within a given interval of unit length. We find sharp upper and lower bounds for this denominator, and observe seemingly chaotic fluctuations between those bounds.


Surjectivity of Maps Induced on Matrices by Polynomials and Entire Functions

p. 260.

Shubhodip Mondal

We determine a necessary and sufficient condition for a polynomial over an algebraically closed field k to induce a surjective map on matrix algebras Mn(k) for n ≥ 2. The criterion is given in terms of algebraic conditions on the polynomial and the proof uses simple linear algebra. Following that, we formulate and prove a corresponding result for entire functions as well.


A New Regular (Compound) Polyhedron (of Infinite Kepler-Poinsot Type)

p. 265.

Dirk Huylebrouck

The five Platonic polyhedra combine identical regular polygons such that the spatial angles between them are identical. Kepler and later Poinsot added four more regular polyhedra by allowing star polygons or self-intersecting faces. Petrie and Coxeter added three more using infinitely repeating elements. In the present note we combine the Kepler–Poinsot generalization with the Petrie–Coxeter generalization to obtain a new regular (compound) polyhedron.


A Proof of Bonnet's Version of the Mean Value Theorem by Methods of Cauchy

p. 269.

Joseph Plante

The proof of the mean value theorem for differentiable functions presented in modern calculus texts is due to Bonnet (1860s) and depends in an essential way on the extreme value property for continuous functions proved by Weierstrass (1861). This proof of the mean value theorem has intuitive quality but mainstream texts omit the proof of Weierstrass’s result on account of its difficulty, thereby depriving the reader of a complete proof of the mean value theorem. Forty years earlier, Cauchy proved the intermediate value property for continuous functions and made a significant but flawed effort to prove a mean value inequality. This note presents a proof of the modern mean value theorem using methods of Cauchy. The key geometrical insight is provided by a simple inequality involving the slopes of sides of a triangle whose vertices lie in the graph of the function. This inequality motivates the use of the intermediate value property in the proof of the mean value theorem and facilitates final determination of the required value of the derivative. An advantage of this alternative to Bonnet’s proof is that it is based directly on the completeness of the real numbers rather than on the result of Weierstrass.


Problems and Solutions

p. 274.


Book Review

p. 282.

Summing it Up: From One Plus One to Modern Number Theory by Avner Ash and Robert Gross

Reviewed by Dominic Lanphier



100 Years Ago This Month in The American Mathematical Monthly

p. 240.