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Mathematics Magazine - April 2018

Mathematics Magazine April 2018 Cover

In the first article, William Dunham reviews Möbius’ use in analysis of his namesake function and Euler’s anticipation of the function. In the next article, Tom Edgar considers visually appealing dissection proofs of staircase series. Francisco Sánchez and José María Sanchis follow by applying Darboux sums to sum the alternating harmonic series.

Two articles on group theory are next. Chase Saucier uses a combinatorial approach to prove Wilson’s theorem for finite Abelian groups, and Robert Heffernan, Des MacHale, and Brendan McCann are motivated by Cayley’s theorem to ask five elementary questions about embeddings of finite groups, which they answer for groups of order 15 or less.

Using multivariable calculus, Joseph Previte and Michelle Previte show that most triangles in n-dimensional space are acute for n ≥ 3. For triangles in the plane, Stefan Catiou and Allan Berrelle introduce a new curve which is the envelope of all lines that bisect the triangle’s perimeter.

Michael Brilleslyper and Lisbeth Schaubroeck consider a simple family of trinomials the roots of which, when plotted in the complex plane, form intriguing patterns. They conjecture a formula regarding the number of roots which lie inside the unit circle, then prove the conjecture for a special case.

There are a number of shorter articles, too. Further, as with every issue in 2018, there is a Partiti puzzle (see the February issue for an introduction). The popular Problems and Solutions and the Reviews complete the issue.

Michael A. Jones, Editor


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Vol. 91, No. 2, pp. 81 – 160


Letter from the Editor

p. 82.

Michael A. Jones

DOI: 10.1080/0025570X.2018.1432929

The Early (and Peculiar) History of the Möbius Function

p. 83.

William Dunham

The Möbius function is a fixture of modern courses in number theory. It is usually traced back to an 1832 paper by August Ferdinand Möbius where the function unexpectedly arose in answer to an analytic, rather than a number theoretic, question. But perhaps more unexpected is that the function can be found in Leonhard Euler's classic text, Introductio in analysis infinitorum, from 1748. Besides presenting the origins of what might be called the “Euler/ Möbius” function, this article is a reminder that the history of mathematics holds its share of surprises.

DOI: 10.1080/0025570X.2017.1413921

Partiti Puzzle

p. 91.

Lai Van Duc Thinh

DOI: 10.1080/0025570X.2017.1421371

Staircase Series

p. 92.

Tom Edgar

We provide a wordless calculation of the series ∑i = 0 k2rk for any 0 < r < 1. We extend this visual proof to give an algebraic reduction formula for ∑i = 0 knrk where n ⩾ 1 is an integer, and we discuss a combinatorial consequence of this formula.

DOI: 10.1080/0025570X.2017.1415584

Darboux Sums and the Sum of the Alternating Harmonic Series

p. 96.

Francisco Sánchez & José María Sanchis

We provide a simple proof that the alternating harmonic series sums to the natural logarithm of 2 by expressing partial sums as a Darboux sum of a function on [0, 1] and on the computation of a limit as an integral.

DOI: 10.1080/0025570X.2017.1408380

A Combinatorial Approach to Wilson’s Theorem for Finite Abelian Groups

p. 97.

Chase Saucier

The generalized pigeonhole principle is used to deduce an analog of Wilson’s theorem for finite Abelian groups. An involutive approach and an approach using Euler’s congruence theorem are also discussed. Other generalizations of Wilson’s theorem are presented as well. We conclude with a discussion of how one might apply the generalized pigeonhole principle in the non-Abelian case.

DOI: 10.1080/0025570X.2017.1419748

Cayley’s Theorem Revisited: Embeddings of Small Finite Groups

p. 103.

Robert Heffernan, Des MacHale & Brendan McCann

Motivated by Cayley’s theorem, we pose five elementary questions concerning embeddings of finite groups and answer them for groups of order 15 or less. Most of our results are easily verified. They provide a novel perspective on group theory that has its roots in the historical origins of the subject.

DOI: 10.1080/0025570X.2017.1419770

Some Taylor Series without Taylor's Theorem

p. 112.

Mark Dalthorp

We present an elementary derivation of the Taylor series of ex, sin  x, and cos   x.

DOI: 10.1080/0025570X.2017.1408980

Triangles in Wonderland

p. 113.

Joseph P. Previte & Michelle Previte

In this paper, we deduce that most triangles in n-dimensional Euclidean space are acute for n ⩾ 3, unlike the case in the plane which was originally considered by Lewis Carroll in 1893 and further developed in this Magazine by Richard Guy in 1993. Further, we produce a formula that computes the percentage of obtuse triangles in any dimension n ⩾ 3 using generalized spherical coordinates. These results are novel for n ⩾ 3 and are attainable by standard techniques taught in a typical undergraduate multivariable calculus course.

DOI: 10.1080/0025570X.2017.1419769

Bisecting the Perimeter of a Triangle

p. 121.

Allan Berele & Stefan Catoiu

We introduce a new curve: the “perimeter-bisecting deltoid of a triangle” is the envelope of all lines that bisect its perimeter. This is a six-sided curve in the shape of the Greek letter delta consisting of three line segments and three segments of parabolas. We describe this curve both as analytic and as geometric locus, compute the area enclosed by it, and classify the points of the triangle according to the number of distinct perimeter-bisecting lines through them.

DOI: 10.1080/0025570X.2017.1418589

The Reciprocal Fibonacci Function

p. 134.

Michael Lord

Properties of the power series whose coefficients are the reciprocals of the Fibonacci numbers are derived. It is shown that this function is equivalent to a generalized Lambert series. An application of the Rodgers–Fine identity provides a fast convergent series representation of the function.

DOI: 10.1080/0025570X.2018.1426946

A Note on the AM-GM Inequality

p. 138.

József Sándor

We show that, a generalization of the AM-GM inequality, proved by Lagrange multipliers, can be proved by an elementary method.

DOI: 10.1080/0025570X.2017.1409533

Polar Bear

p. 139.

Robert Haas

DOI: 10.1080/0025570X.2017.1415574

Even Perfect Numbers End in 6 or 28

p. 140.

Roger B. Nelsen

We partition triangular numbers to show that even perfect numbers end in 6 or 28.

DOI: 10.1080/0025570X.2017.1409558

Solution to the Partiti Puzzle

p. 141.

Lai Van Duc Thinh

DOI: 10.1080/0025570X.2017.1421372

Counting Interior Roots of Trinomials

p. 142.

Michael A. Brilleslyper & Lisbeth E. Schaubroeck

We consider a simple family of trinomials. The roots of this family, when plotted in the complex plane, form intriguing patterns which motivate some natural questions about their distribution. In particular, we conjecture a formula that counts the number of roots that lie inside the unit circle. We provide a proof of this conjecture for a special case. The proof makes use of basic geometry, number theoretic calculations, and an application of Rouché’s theorem. The result partially answers a previous open question from an earlier paper that investigated when this family of trinomials has roots on the unit circle.

DOI: 10.1080/0025570X.2017.1420332

Problems and Solutions

p. 151.

Proposals, 2041-2045

Quickies, 1079-1080

Solutions, 2016-2020

Answers, A1079-A1080

DOI: 10.1080/0025570X.2018.1429759


p. 159.

Loving mathematics; integration; credit scoring; faster MRIs; being gifted

DOI: 10.1080/0025570X.2018.1432204