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E-raamat: Biscuits of Number Theory

Edited by (Virginia Polytechnic Institute and State University), Edited by (Harvey Mudd College, California)
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An anthology of articles designed to supplement a first course in number theory.

Brown (mathematics, Virginia Tech) and Benjamin (mathematics, Harvey Mudd College) collect award-winning articles that can be appreciated by anyone who has taken, or is taking, a first course in number theory. Why call them biscuits? According to the editors, "Each item is not too big, easily digested, and makes you feel all warm and fuzzy when you're through." The articles were previously published in various mathematical journals over the past 25 years, with a few articles going back to the 1950s through the 1970s. Articles are grouped in sections on arithmetic, primes, irrationality and continued fractions, sums of squares and polygonal numbers, Fibonacci numbers, number-theoretic functions, and elliptic curves, cubes, and Fermat's last theorem. Many of the articles conclude with 'second helpings,' references leading students to related topics. The articles can be used as starting points for discussions, follow-ups to classroom presentations, and research projects. The book can be used as a supplement for a number theory course, especially one that requires students to write papers or do outside reading. There is no subject index. Annotation ©2009 Book News, Inc., Portland, OR (booknews.com)

In Biscuits of Number Theory, the editors have chosen articles that are exceptionally well written and that can be appreciated by anyone who has taken (or is taking) a first course in number theory. This book could be used as a textbook supplement for a number theory course, especially one that requires students to write papers or do outside reading. Here are some of the possibilities: The collection is divided into seven chapters: Arithmetic, Primes, Irrationality, Sums of Squares and Polygonal Numbers, Fibonacci Numbers, Number Theoretic Functions, and Elliptic Curves, Cubes, and Fermat's Last Theorem. As with any anthology, you don't have to read the Biscuits in order. Dip into them anywhere: pick something from the Table of Contents that strikes your fancy, and have at it. If the end of an article leaves you wondering what happens next, then by all means dive in and do some research. You just might discover something new!

A collection of articles on number theory that are divided into seven main parts: Arithmetic, Primes, Irrationality, Sums of Squares and Polygonal Numbers, Fibonacci Numbers, Number Theoretic Functions, and Elliptic Curves, Cubes, and Fermat's Last Theorem.
Introduction xi
Part I: Arithmetic
1(58)
A Dozen Questions About the Powers of Two
3(10)
James Tanton
From 30 to 60 is Not Twice as Hard
13(4)
Michael Dalezman
Reducing the Sum of Two Fractions
17(6)
Harris S. Shultz
Ray C. Shiflett
A Postmodern View of Fractions and Reciprocals of Fermat Primes
23(16)
Rafe Jones
Jan Pearce
Visible Structures in Number Theory
39(14)
Peter Borwein
Loki Jorgenson
Visual Gems of Number Theory
53(6)
Roger B. Nelsen
Part II: Primes
59(46)
A New Proof of Euclid's Theorem
61(2)
Filip Saidak
On the Infinitude of the Primes
63(2)
Harry Furstenberg
On the Series of Prime Reciprocals
65(2)
James A. Clarkson
Applications of a Simple Counting Technique
67(2)
Melvin Hausner
On Weird and Pseudoperfect Numbers
69(8)
S. J. Benkoski
P. Erdos
A Heuristic for the Prime Number Theorem
77(8)
Hugh L. Montgromery
Stan Wagon
A Tale of Two Sieves
85(20)
Carl Pomerance
Part III: Irrationality and Continued Fractions
105(36)
Irrationality of the Square Root of Two---A Geometric Proof
107(2)
Tom M. Apostol
Math Bite: Irrationality of √m
109(2)
Harley Flanders
A Simple Proof that π is Irrational
111(2)
Ivan Niven
r, e and Other Irrational Numbers
113(2)
Alan E. Parks
A Short Proof of the Simple Continued Fraction of e
115(6)
Henry Cohn
Diophantine Olympics and World Champions: Polynomials and Primes Down Under
121(8)
Edward B. Burger
An Elementary Proof of the Wallis Product Formula for Pi
129(4)
Johan Wastlund
The Orchard Problem
133(8)
Ross Honsberger
Part IV: Sums of Squares and Polygonal Numbers
141(14)
A One-Sentence Proof that every Prime p ≡ 1 (mod 4) is a Sum of Two Squares
143(2)
D. Zagier
Sum of Squares II
145(2)
Martin Gardner
Dan Kalman
Sums of Squares VIII
147(2)
Roger B. Nelsen
A Short Proof of Cauchy's Polygonal Number Theorem
149(4)
Melvyn B. Nathanson
Genealogy of Pythagorean Triads
153(2)
A. Hall
Part V: Fibonacci Numbers
155(40)
A Dozen Questions About Fibonacci Numbers
157(10)
James Tanton
The Fibonacci Numbers---Exposed
167(16)
Dan Kalman
Robert Mena
The Fibonacci Numbers---Exposed More Discretely
183(12)
Arthur T. Benjamin
Jennifer J. Quinn
Part VI: Number-Theoretic Functions
195(60)
Great Moments of the Riemann zeta Function
199(18)
Jennifer Beineke
Chris Hughes
The Collatz Chameleon
217(6)
Marc Chamberland
Bijecting Euler's Partition Recurrence
223(2)
David M. Bressoud
Doron Zeilberger
Discovery of a Most Extraordinary Law of the Numbers Concerning the Sum of Their Divisors
225(8)
Leonard Euler
George Polya
The Factorial Function and Generalizations
233(18)
Manjul Bhargava
An Elementary Proof of the Quadratic Reciprocity Law
251(4)
Sey Y. Kim
Part VII: Elliptic Curves, Cubes and Fermat's Last Theorem
255(56)
Proof Without Words: Cubes and Squares
257(2)
J. Barry Love
Taxicabs and Sums of Two Cubes
259(14)
Joseph H. Silverman
Three Fermat Trails to Elliptic Curves
273(12)
Ezra Brown
Fermat's Last Theorem in Combinatorial Form
285(2)
W.V. Quine
``A Marvelous Proof''
287(24)
Fernando Q. Gouvea
About the Editors 311
Arthur Benjamin earned his B.S. in Applied Mathematics from Carnegie Mellon and his Ph.D. in Mathematical Sciences from Johns Hopkins. Since 1989, he has taught at Harvey Mudd College, where he is Professor of Mathematics and past Chair. In 2000, he received the Haimo Award for Distinguished Teaching from the Mathematical Association of America. Since 2006 he has served as the MAA's Polya Lecturer. He has been featured in numerous magazines, television and radio programmes. Ezra Brown grew up in New Orleans and has degrees from Rice University and Louisiana State University. Since 1969 he has been in the Mathematics Department at Virginia Tech, where he is currently Alumni Distinguished Professor. He is the author of some sixty papers, mostly in number theory and discrete mathematics. He received the Outstanding Teacher Award from the MD/DC/VA Section of the MAA, and he currently serves as that section's governor. He received the Carl Allendoerfer Award (2003) and three George Polya Awards (2000, 2001, 2006) from the MAA for expository writing.