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College Mathematics Journal Contents—March 2017

Among the articles in this issue of The College Mathematics Journal are several that make extensive and essential use of graphics. The most dramatic is the alliterative "A Visual Validation of Viète's Verification" with several outputs from Sage's complex_plot command showing all roots of various polynomials. The Chinese remainder clock uses five circles to tell time with a favorite theorem of number theory and, amazingly, one can use a measurement from an image on the computer screen to make a good prediction of the International Space Station's orbit. In other articles, the perennially popular Pythagorean triples and Fibonacci numbers make appearances, along with a thoughtful consideration of the subtleties that arise when dealing with factorization. —Brian Hopkins

Vol. 48, No. 2, pp. 81-160


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The Chinese Remainder Clock

p. 82.

Antonella Perucca

We present an analog clock with five hands that illustrates the Chinese remainder theorem and that can be understood also by nonmathematicians. Moreover, we interpret the Chinese remainder theorem in terms of rotations and prove it without equations.

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A Visual Validation of Viète’s Verification

p. 90.

Tom Edgar and N. Chris Meyer

We visually investigate Viète’s method for funding for the roots of the depressed cubic equation using transformations and complex plots from Sage to consider the three cases

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Integral Values of the Generating Functions of Fibonacci and Lucas Numbers

p. 97.

Prapanpong Pongsriiam

Hong recently explored when the value of the generating function of the Fibonacci sequence is an integer. He noticed a pattern and raised some questions about it. Here, we completely answer those questions.

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Computing a Satellite Orbit From Photographs

p. 102.

Donald Teets

Simple measurements are taken directly from a photograph showing the track of the International Space Station over a 20-second interval. From these measurements, we are able to compute parameters for a circular approximation of the space station’s orbit and show that they compare favorably to NASA values for the same quantities.

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Forgotten Equations

p. 111.

Kazim Ali

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Pythagorean Triples for Easy Solutions of Certain Quadratics and a Newly Generated Tree

p. 112.

Edward R. Forringer

Certain quadratic equations can be solved by inspection if they share a pattern of differences with a Pythagorean triple. The connection between Pythagorean triples and quadratic equations also generates a tree of Pythagorean triples that has been discussed before but generated using matrices.

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Pythagorean Triples, Complex Numbers, and Perplex Numbers

p. 115.

Howard Sporn

After reviewing the use of Gaussian integers to generate Pythagorean triples, we consider the history of the perplex numbers and use them in an analogous way to generate Pythagorean triples. We also show that primitive Pythagorean triples form a group under the perplex number system, different from the group determined by the complex number representations.

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If a Prime Divides a Product…

p. 123.

Steven J. Miller and Cesar E. Silva

One of the greatest difficulties encountered by all in their first proof-intensive class is subtly assuming an unproven fact in a proof. The purpose of this note is to describe a specific instance where this can occur, namely in results related to unique factorization and the concept of the greatest common divisor.

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On the Arithmetic Mean of the Square Roots of the First n Positive Integers

p. 129.

Mircea Merca

In the paper, the author presents an asymptotic formula for the arithmetic mean of the square roots of the first n positive integers using only tools from undergraduate calculus.

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Classroom Capsules

Homographic Pencils for the Ellipse and the Hyperbola

p. 134.

Francisco Javier García Capitán

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A Very Short Proof that the Sum of Independent Normal Random Variables Is Normal

p. 137.

Bennett Eisenberg

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Problems and Solutions

p. 138.

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Book Reviews

Some Applications of Geometric Thinking and Moving Things Around  By Bowen Kerins, Darryl Yong, Al Cuoco, Glenn Stevens, and Mary Pilgrim

p. 146.

Reviewed by: Thomas Dick

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Media Highlights

p. 153.

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