**Dolciani Mathematical Expositions**

Volume: 57;
2021;
207 pp;
Softcover

MSC: Primary 11; 14;

**Print ISBN: 978-1-4704-5622-1
Product Code: DOL/57**

List Price: $59.00

AMS Member Price: $44.25

MAA Member Price: $44.25

**Electronic ISBN: 978-1-4704-6502-5
Product Code: DOL/57.E**

List Price: $59.00

AMS Member Price: $44.25

MAA Member Price: $44.25

#### Supplemental Materials

# A Gateway to Number Theory: Applying the Power of Algebraic Curves

Share this page
*Keith Kendig*

MAA Press: An Imprint of the American Mathematical Society

Challenge: Can you find all the integers \(a\),
\(b\), \(c\) satisfying \(2a^{2} + 3b^{2} =
5c^{2}\)? Looks simple, and there are in fact a number of easy
solutions. But most of them turn out to be anything but obvious! There
are infinitely many possibilities, and as any computer will tell you,
each of \(a\), \(b\), \(c\) will usually be
large. So the challenge remains …

Find all integers \(a\), \(b\), \(c\)
satisfying \(2a^{2} + 3b^{2} = 5c^{2}\).

A major advance in number theory means this book can give an easy
answer to this and countless similar questions. The idea behind the
approach is transforming a degree-two equation in integer variables
\(a\), \(b\), \(c\) into a plane curve defined by
a polynomial. Working with the curve makes obtaining solutions far
easier, and the geometric solutions then get translated back into
integers. This method morphs hard problems into routine ones and
typically requires no more than high school math. (The complete
solution to \(2a^{2} + 3b^{2} = 5c^{2}\) is included in the
book.)

In addition to equations of degree two, the book addresses
degree-three equations—a branch of number theory that is today
something of a cottage industry, and these problems translate into
“elliptic curves”. This important part of the book
includes many pictures along with the exposition, making the material
meaningful and easy to grasp.

This book will fit nicely into an introductory course on number
theory. In addition, the many solved examples, illustrations, and
exercises make self-studying the book an option for students, thus
becoming a natural candidate for a capstone course.

#### Readership

Undergraduate and graduate students interested in elliptic curves.

#### Table of Contents

# Table of Contents

## A Gateway to Number Theory: Applying the Power of Algebraic Curves

- Cover Cover11
- Title page iii5
- Copyright iv6
- Contents vii9
- Preface xi13
- Chapter 1. A Marriage for the Ages 119
- 1.1. Mathematics and the Hawaiian Islands 119
- 1.2. A Model 220
- 1.3. Appearance of the Rational Unit Circle, Part I 321
- 1.4. Appearance of the Rational Unit Circle, Part II 826
- 1.5. What About Other Rational Circles? 927
- 1.6. A Parametric Formula for the Unit Circle in ℚ² 1230
- 1.7. More General Quadratic Problems 1533
- 1.8. Conics 2139
- 1.9. Our Method Also Works in Reverse 2644
- 1.10. Surveying All Solutions 2947
- 1.11. The Discriminant 3048
- 1.12. Finding a Rational Point 3351
- 1.13. What About Degree 1? 3452

- Chapter 2. Viewing the Whole Algebraic Curve 3755
- Chapter 3. Entering the World of Elliptic Curves 5573
- Chapter 4. Every Elliptic Curve Is a Group! 8199
- Chapter 5. A Million-Dollar Challenge 105123
- 5.1. Breaking Up a Task into Many Smaller Ones 105123
- 5.2. The Birch and Swinnerton-Dyer Conjecture 110128
- 5.3. The Notion of Expected Rank 119137
- 5.4. Expected Rank of a Random Elliptic Curve 120138
- 5.5. The Tale of Average Rank 122140
- 5.6. Rank Results Without GRH or BSD 127145
- 5.7. About Manjul Bhargava 130148

- Chapter 6. Every Real Elliptic Curve Lives in a Donut 137155
- Chapter 7. The Genus 155173
- Chapter 8. In Conclusion . . . 165183
- Appendix A. What Is a Smooth Complex Curve? 171189
- Appendix B. Algebraic Curves in the Disk Model 181199
- Appendix C. Some Code for This Book’s Programs 193211
- Bibliography 201219
- Index 205223
- Back Cover Back Cover1226