The November *Monthly* opens with a tribute to the late Monthly Associate Editor Jonathan M. Borwein, who authored 25 plus Monthly articles written over a span of more than 30 years.

If you like to factor, check out "Finding Elements With Given Factorization Lengths and Multiplicities," by Paul Baginski, Ryan Rodriguez, George J. Schaeffer, and Yiwei She. This issue also has examples of non-Euclidean PIDs, from Anthony J. Bevelacqua's Note, "A Family of Non-Euclidean PIDs."

Gerald Folland reviews *The Real and the Complex: A History of Analysis in the 19th Century* by Jeremy Gray and our Problems and Solutions section will keep you busy over your Thanksgiving break. Stay tuned for December when we learn about "Sets of Lengths" from Alfred Geroldinger.

— *Scott T. Chapman, Editor*

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## Table of Contents

### A Letter From the Editor: Jonathan M. Borwein (1951–2016)

p. 847.

Scott T. Chapman

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.9.847

### Finding Elements With Given Factorization Lengths and Multiplicities

p. 849.

Paul Baginski, Ryan Rodriguez, George J. Schaeffer, and Yiwei She

Many algebraic number rings exhibit nonunique factorization of elements into irreducibles. Not only can the irreducibles in the factorizations be different, but the *number* of irreducibles in the factorizations can also vary. A basic question then is: Which sets can occur as the set of factorization lengths of an element? Moreover, how often can each factorization length occur? While these questions are most pertinent in algebraic number rings, their pertinence extends to Dedekind domains and a broader class of structures called Krull monoids. Surprisingly, for a large subclass of Krull monoids, Kainrath was able to resolve completely the question of which length sets and length multiplicities can be realized. In this article, we explain the context of Kainrath's theorem and give a constructive proof for an important case, namely Krull monoids with infinite nontorsion class group. We also construct length sets in a case not covered by Kainrath's theorem to illustrate the difficulty of the general problem.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.9.849

### Stråhle's Equal-Tempered Triumph

p. 871.

Lyle Ramshaw

In 1743, the Swedish organ builder Daniel P. Stråhle gave an elegant geometric construction for determining the precise pitches of musical notes—for example, for locating frets along the neck of a guitar. We analyze how closely Stråhle's construction approximates equal temperament as a function of the *pitch gamut*, that is, the frequency range covered, and the *tilt ratio* of the perspectivity that locates the frets. With a pitch gamut of one octave, the tilt ratio then needing to be roughly , we show that Stråhle's choice of the ratio 24/17 gives better fretboards than the continued-fraction convergent 17/12.We also compare the accuracy of Stråhle's construction to that of its main competitor, a saddle-shifting variant of Vincenzo Galilei's* rule of 18*. Galilei is more accurate on a gamut of one octave, but Stråhle is more accurate on the quarter-octave gamut that would tempt a compass-equipped perfectionist.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.9.871

### A Note on Harmonic Functions on Surfaces

p. 884.

Jean C. Cortissoz

We review and give elementary proofs of Liouville-type properties of harmonic and subharmonic functions in the plane endowed with a complete Riemannian metric, and prove a gap theorem for the possible growth of harmonic functions when this metric has nonnegative Gaussian curvature.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.9.884

### Another Memorable Mongolian Olympiad Problem: A Two-Colored Rectangular Lattice

p. 894.

Uuganbaatar Ninjbat

By proving the following result, we provide a complete solution to an open problem which appeared in the last Mongolian Mathematical Olympiad. Suppose the cells of a finite rectangular lattice are colored with white or black such that each black cell has an even number of white neighbors, i.e., white cells with a common side with the black cell. Then, the white cells can be partitioned (or colored) into two groups such that for each black cell, its white neighbors are divided evenly between them.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.9.894

### The Multistep Friendship Paradox

p. 900.

Josh Brown Kramer, Jonathan Cutler and A. J. Radcliffe

The friendship paradox, proved by Feld in 1991, states that “on average, your friends have more friends than you do.” In fact, Feld proved two versions of the paradox. We discuss generalizations of each of them that talk about the average number of friends that, for instance, a friend of a friend of a friend of a friend has.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.9.900

### The Volume of the Spatial Region Corresponding to *n* × *n* Correlation Matrices

p. 909.

Sean Eastman, Selwyn Hollis, Kawee Numpacharoen and Jared Schlieper

A correlation matrix may be associated with a point in a hypercube of a certain dimension, each of whose coordinates has magnitude less than or equal to one. Using a spherical form of the Cholesky decomposition, we compute the volume of the subset of the hypercube corresponding to all valid correlation matrices of a given size. Doing so enables us to determine the probability that a randomly chosen point in the hypercube corresponds to a valid correlation matrix.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.9.909

## Notes

### Primitivity and Orbit Decomposition

p. 920.

Bernd Schomburg

We give sufficient conditions for the primitivity of a transitive permutation group in terms of the orbit structure of a subgroup.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.9.920

### On the Success of Mishandling Euclid's Lemma

p. 924.

Adrian W. Dudek

We examine Euclid’s lemma that if *p* is a prime number such that *p*|*ab*, then *p* divides at least one of *a* or *b*. Specifically, we consider the common misapplication of this lemma to numbers that are not prime, as is often made by undergraduate students. We show that a randomly chosen implication of the form *r*|*ab* ⇒ *r*|*a* or* r*|*b* is *almost surely* false in a probabilistic sense, and we quantify this with a corresponding asymptotic formula.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.9.924

### A Simple Proof of a Generalized Cauchy–Binet Theorem

p. 928.

Alan J. Hoffman and Chai Wah Wu

The Cauchy–Binet theorem is the main step in proving the multiplicative property of compound matrices. We present here a novel elementary proof of a generalization of the Cauchy–Binet theorem using properties of square matrices.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.9.928

### Optimal Bounds for the Roots of Polynomials

p. 931.

Nicolae Anghel

A general criterion for detecting optimal bounds on the roots of univariate polynomials with complex coefficients is given. The criterion is then applied to three such bounds, two of them being the classical Cauchy and Lagrange bounds.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.9.931

### A Family of Non-Euclidean PIDs

p. 936.

Anthony J. Bevelacqua

Using elementary methods we construct a family of non-Euclidean principal ideal domains.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.9.936

## Problems and Solutions

p. 941.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.123.9.941

## Book Review

p. 949.

*The Real and the Complex: A History of Analysis in the 19th Century* by Jeremy Gray

Reviewed by Gerald B. Folland

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.123.9.949

## MathBits

### Two Extensions of the Sury's Identity

p. 919.

Ivica Martinjak

### An Alternate Proof of the Binomial Theorem

p. 940.

Kuldeep Kumar Kataria