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Biscuits of Number Theory

Edited by: Arthur T. Benjamin
MAA Press: An Imprint of the American Mathematical Society
Available Formats:
Electronic ISBN: 978-1-4704-5843-0
Product Code: DOL/34.E
List Price: $64.00 MAA Member Price:$48.00
AMS Member Price: $48.00 Click above image for expanded view Biscuits of Number Theory Edited by: Arthur T. Benjamin MAA Press: An Imprint of the American Mathematical Society Available Formats:  Electronic ISBN: 978-1-4704-5843-0 Product Code: DOL/34.E  List Price:$64.00 MAA Member Price: $48.00 AMS Member Price:$48.00
• Book Details

Dolciani Mathematical Expositions
Volume: 342009; 311 pp

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.

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; 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!

• 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 One-Sentence 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: Number-Theoretic 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”

• Reviews

• A collection of accessible and even profound essays on number theory gleaned from a wide variety of writers and journals--everyone 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
• Requests

Review Copy – for reviewers who would like to review an AMS book
Accessibility – to request an alternate format of an AMS title
Volume: 342009; 311 pp

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.

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; 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!

• 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 One-Sentence 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: Number-Theoretic 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 journals--everyone 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
Review Copy – for reviewers who would like to review an AMS book
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.