eBookISBN:  9781470458430 
Product Code:  DOL/34.E 
List Price:  $64.00 
MAA Member Price:  $48.00 
AMS Member Price:  $48.00 
eBook ISBN:  9781470458430 
Product Code:  DOL/34.E 
List Price:  $64.00 
MAA Member Price:  $48.00 
AMS Member Price:  $48.00 

Book DetailsDolciani Mathematical ExpositionsVolume: 34; 2009; 311 pp
In Biscuits of Number Theory, the editors have chosen articles that are exceptionally wellwritten 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.
The editors give examples of some of the possibilities. The collection is divided into seven chapters: Arithmetic; Primes; Irrationality and Continued Fractions; Sums of Squares and Polygonal Numbers; Fibonacci Numbers; NumberTheoretic 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! 
Table of Contents

Articles

Part I: Arithmetic

James Tanton  A Dozen Questions About the Powers of Two

Michael Dalezman  From 30 to 60 is Not Twice as Hard

Harris Shultz and Ray C. Shiflett  Reducing the Sum of Two Fractions

Rafe Jones and Jan Pearce  A Postmodern View of Fractions and Reciprocals of Fermat Primes

Peter Borwein and Loki Jörgenson  Visible Structures in Number Theory

Roger B. Nelsen  Visual Gems of Number Theory

Part II: Primes

Filip Saidak  A New Proof of Euclid’s Theorem

Harry Furstenberg  On the Infinitude of the Primes

James A. Clarkson  On the Series of Prime Reciprocals

Melvin Hausner  Applications of a Simple Counting Technique

S. J. Benkoski and P. Erdős  On Weird and Pseudoperfect Numbers

Hugh L. Montgromery and Stan Wagon  A Heuristic for the Prime Number Theorem

Carl Pomerance  A Tale of Two Sieves

Part III: Irrationality and Continued Fractions

Tom M. Apostol  Irrationality of the Square Root of Two—A Geometric Proof

Harley Flanders  Math Bite: Irrationality of $\sqrt {m}$

Ivan Niven  A Simple Proof that $\pi $ is irrational

Alan E. Parks  $\pi , e$ and Other Irrational Numbers

Henry Cohn  A Short Proof of the Simple Continued Fraction of $e$

Edward B. Burger  Diophantine Olympics and World Champions: Polynomials and Primes Down Under

Johan Wästlund  An Elementary Proof of the Wallis Product Formula for Pi

Ross Honsberger  The Orchard Problem

Part IV: Sums of Squares and Polygonal Numbers

D. Zagier  A OneSentence Proof that every Prime $p\equiv 1$ (mod 4) is a Sum of Two Squares

Martin Gardner and Dan Kalman  Sum of Squares II

Roger B. Nelsen  Sum of Squares VIII

Melvyn B. Nathanson  A Short Proof of Cauchy’s Polygonal Number Theorem

A. Hall  Genealogy of Pythagorean Triads

Part V: Fibonacci Numbers

James Tanton  A Dozen Questions About Fibonacci Numbers

Dan Kalman and Robert Mena  The Fibonacci Numbers—Exposed

Arthur T. Benjamin and Jennifer J. Quinn  The Fibonacci Numbers—Exposed More Discretely

Part VI: NumberTheoretic Functions

Jennifer Beineke and Chris Hughes  Great Moments of the Riemann zeta Function

Marc Chamberland  The Collatz Chameleon

David M. Bressoud and Doron Zeilberger  Bijecting Euler’s Partition Recurrence

Leonard Euler and Translated by George Pólya  Discovery of a Most Extraordinary Law of the Numbers Concerning the Sum of Their Divisors

Manjul Bhargava  The Factorial Function and Generalizations

Sey Y. Kim  An Elementary Proof of the Quadratic Reciprocity Law

Part VII: Elliptic Curves, Cubes and Fermat’s Last Theorem

J. Barry Love  Proof Without Words: Cubes and Squares

Joseph H. Silverman  Taxicabs and Sums of Two Cubes

Ezra Brown  Three Fermat Trails to Elliptic Curves

W. V. Quine  Fermat’s Last Theorem in Combinatorial Form

Fernando Q. Gouvêa  “A Marvelous Proof”


Additional Material

Reviews

A collection of accessible and even profound essays on number theory gleaned from a wide variety of writers and journalseveryone from Euler to Quine, plus many recent popular expositions. An invigorating and generally undemanding excursion into surprise. A first rate book.
Bob Lockhart, London Math Society Newsletter 
The authors represented include some of the best expositors of elementary number theory: Peter Borwein, Stan Wagon, Carl Pomerance, Ivan Niven, Edward Berger, Ross Honsberger, and Martin Gardent, just to name a few. … it's good when a book has some content above the level of the typical reader, because this will intrigue some readers sufficiently that they'll feel the need to learn the required material. The challenge is to have the right amount, and my feeling is that this book has a good balance of material.
Jeffrey Shallit, Sigact News


RequestsReview Copy – for reviewers who would like to review an AMS bookAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Additional Material
 Reviews
 Requests
In Biscuits of Number Theory, the editors have chosen articles that are exceptionally wellwritten 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.
The editors give examples of some of the possibilities. The collection is divided into seven chapters: Arithmetic; Primes; Irrationality and Continued Fractions; Sums of Squares and Polygonal Numbers; Fibonacci Numbers; NumberTheoretic 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!

Articles

Part I: Arithmetic

James Tanton  A Dozen Questions About the Powers of Two

Michael Dalezman  From 30 to 60 is Not Twice as Hard

Harris Shultz and Ray C. Shiflett  Reducing the Sum of Two Fractions

Rafe Jones and Jan Pearce  A Postmodern View of Fractions and Reciprocals of Fermat Primes

Peter Borwein and Loki Jörgenson  Visible Structures in Number Theory

Roger B. Nelsen  Visual Gems of Number Theory

Part II: Primes

Filip Saidak  A New Proof of Euclid’s Theorem

Harry Furstenberg  On the Infinitude of the Primes

James A. Clarkson  On the Series of Prime Reciprocals

Melvin Hausner  Applications of a Simple Counting Technique

S. J. Benkoski and P. Erdős  On Weird and Pseudoperfect Numbers

Hugh L. Montgromery and Stan Wagon  A Heuristic for the Prime Number Theorem

Carl Pomerance  A Tale of Two Sieves

Part III: Irrationality and Continued Fractions

Tom M. Apostol  Irrationality of the Square Root of Two—A Geometric Proof

Harley Flanders  Math Bite: Irrationality of $\sqrt {m}$

Ivan Niven  A Simple Proof that $\pi $ is irrational

Alan E. Parks  $\pi , e$ and Other Irrational Numbers

Henry Cohn  A Short Proof of the Simple Continued Fraction of $e$

Edward B. Burger  Diophantine Olympics and World Champions: Polynomials and Primes Down Under

Johan Wästlund  An Elementary Proof of the Wallis Product Formula for Pi

Ross Honsberger  The Orchard Problem

Part IV: Sums of Squares and Polygonal Numbers

D. Zagier  A OneSentence Proof that every Prime $p\equiv 1$ (mod 4) is a Sum of Two Squares

Martin Gardner and Dan Kalman  Sum of Squares II

Roger B. Nelsen  Sum of Squares VIII

Melvyn B. Nathanson  A Short Proof of Cauchy’s Polygonal Number Theorem

A. Hall  Genealogy of Pythagorean Triads

Part V: Fibonacci Numbers

James Tanton  A Dozen Questions About Fibonacci Numbers

Dan Kalman and Robert Mena  The Fibonacci Numbers—Exposed

Arthur T. Benjamin and Jennifer J. Quinn  The Fibonacci Numbers—Exposed More Discretely

Part VI: NumberTheoretic Functions

Jennifer Beineke and Chris Hughes  Great Moments of the Riemann zeta Function

Marc Chamberland  The Collatz Chameleon

David M. Bressoud and Doron Zeilberger  Bijecting Euler’s Partition Recurrence

Leonard Euler and Translated by George Pólya  Discovery of a Most Extraordinary Law of the Numbers Concerning the Sum of Their Divisors

Manjul Bhargava  The Factorial Function and Generalizations

Sey Y. Kim  An Elementary Proof of the Quadratic Reciprocity Law

Part VII: Elliptic Curves, Cubes and Fermat’s Last Theorem

J. Barry Love  Proof Without Words: Cubes and Squares

Joseph H. Silverman  Taxicabs and Sums of Two Cubes

Ezra Brown  Three Fermat Trails to Elliptic Curves

W. V. Quine  Fermat’s Last Theorem in Combinatorial Form

Fernando Q. Gouvêa  “A Marvelous Proof”

A collection of accessible and even profound essays on number theory gleaned from a wide variety of writers and journalseveryone from Euler to Quine, plus many recent popular expositions. An invigorating and generally undemanding excursion into surprise. A first rate book.
Bob Lockhart, London Math Society Newsletter 
The authors represented include some of the best expositors of elementary number theory: Peter Borwein, Stan Wagon, Carl Pomerance, Ivan Niven, Edward Berger, Ross Honsberger, and Martin Gardent, just to name a few. … it's good when a book has some content above the level of the typical reader, because this will intrigue some readers sufficiently that they'll feel the need to learn the required material. The challenge is to have the right amount, and my feeling is that this book has a good balance of material.
Jeffrey Shallit, Sigact News