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MONTHLY, January 2006

**Prime Number Races**

By Andrew Granville and Greg Martin

Andrew@dms.umontreal.ca, gerg@math.ubc.ca

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 4*n*+3 less than *x* then there are of the form 4*n*+1. Similar observations have been made with primes of the form 3*n*+2 and 3*n*+1, with primes of the form 10*n*+3 (or 10*n*+7) and 10*n*+1 (or 10*n*+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

john.hagood@nau.edu, thomson@cs.sfu.ca

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

mfrantz@indiana.edu

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 http://php.indiana.edu/~mathart/animated.

**Notes**

**A Short Proof of the Simple Continued Fraction Expansion of ***e*

by Henry Cohn

cohn@microsoft.com

**A Proof of the Continued Fraction Expansion of e**^{1/M}

by Thomas J. Osler

osler@rowan.edu

**A Continuous Differentiability of Solutions of ODEs with Respect to Initial Conditions**

by Xiongping Dai

xpdai@nju.edu.cn, xiongpingdai@yahoo.com

**Fastest Mixing Markov Chain on a Path**

by Stephen Boyd, Persi Diaconis, Jun Sun, and Lin Xiao

boyd@stanford.edu, sunjun@stanford.edu, lxiao@caltech.edu

**Evolution ofÂ…**

**The Poincaré Conjecture**

by Pawel Strzelecki

**Problems and Solutions**

**Reviews**

**The Pursuit of Perfect Packing **

by Tomaso Aste and Denis Weaire.

Reviewed by Charles Radin

radin@math.utexas.edu

**Kepler’s Conjecture **

by George G. Szpiro

Reviewed by Charles Radin

radin@math.utexas.edu

**Complexities: Women in Mathematics** Edited by Bettye Anne Case and Anne M. Leggett

Reviewed by Shandelle M. Henson

henson@andrews.edu