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American Mathematical Monthly -January 2006



MONTHLY, January 2006

Prime Number Races
By Andrew Granville and Greg Martin
[email protected], [email protected]
This is a survey article on prime number races. Chebyshev noticed in the first half of the nineteenth century that for any given value of x there always seem to be more primes of the form 4n+3 less than x then there are of the form 4n+1. Similar observations have been made with primes of the form 3n+2 and 3n+1, with primes of the form 10n+3 (or 10n+7) and 10n+1 (or 10n+9), and many others besides. More generally, one can consider primes of the form qn+a, qn+b, qn+c, ... for our favorite constants q, a, b, c, ... and try to figure out which forms are "preferred" over the others. In this paper, we describe these phenomena in greater detail and explain the efforts that have been made at understanding them.


Recovering a Function From a Dini Derivative
by John W. Hagood and Brian S. Thomson
[email protected], [email protected]
A function can be recovered up to an additive constant from its derivative, but can the same be said of a single Dini derivative? Dini in 1878 knew that a continuous function was so determined, but he would have had no idea then how to recover the function, having little more than Riemann integration as a tool to work with. A half-century later, Lebesgue addressed the problem bringing in the heavy machinery of measure theory and arcane totalization methods. Our goal is to answer the question using an integral that arises from a Riemann sum process. The investigation is similar in some ways to the case using the derivative, but is more interesting, more varied, and even surprising at points. The path takes us into the integration process of Denjoy and Perron, now known more frequently as the Henstock-Kurzweil integral, and then into a modification of that integral. The presentation is seasoned with historical perspectives.


Some Graphical Solutions of the Kepler Problem
by Marc Frantz
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We typically crown the presentation of calculus by deriving the conic-section shapes of planetary orbits. To avoid spoiling the coronation, we don’t emphasize the fact that we can’t exactly answer the obvious question known as the "Kepler problem." Namely, given a planet's position on the orbit at a given time, exactly where is it at a later time? Although the position cannot be expressed as an elementary function of the time, we can locate the planet at any time using graphs of elementary functions, thereby achieving a partial victory. The history of this endeavor is studded with the names of Kepler himself, John-Dominique Cassini (namesake of the "Cassini division" in the rings of Saturn) and his son Jacques, Johann Franz Encke (namesake of Comet Encke and the "Encke gap" in Saturn’s rings), Christopher Wren, John Wallis, and Isaac Newton. In this article we present a new approach that gives double solutions for both elliptic and hyperbolic orbits, includes both attractive and repulsive accelerations, and leads to interesting animations linking Keplerian motion with other physical processes. Throughout the article we encourage the reader to look at and enjoy these animations, which appear on the website



A Short Proof of the Simple Continued Fraction Expansion of e
by Henry Cohn
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A Proof of the Continued Fraction Expansion of e1/M
by Thomas J. Osler
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A Continuous Differentiability of Solutions of ODEs with Respect to Initial Conditions
by Xiongping Dai
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Fastest Mixing Markov Chain on a Path
by Stephen Boyd, Persi Diaconis, Jun Sun, and Lin Xiao
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Evolution ofÂ…

The Poincaré Conjecture
by Pawel Strzelecki

Problems and Solutions


The Pursuit of Perfect Packing
by Tomaso Aste and Denis Weaire.
Reviewed by Charles Radin
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Kepler’s Conjecture
by George G. Szpiro
Reviewed by Charles Radin
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Complexities: Women in Mathematics Edited by Bettye Anne Case and Anne M. Leggett
Reviewed by Shandelle M. Henson
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