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

Keith Kendig Cleveland State University, Cleveland, OH
MAA Press: An Imprint of the American Mathematical Society
Available Formats:
Softcover ISBN: 978-1-4704-5622-1
Product Code: DOL/57
List Price: $59.00 MAA Member Price:$44.25
AMS Member Price: $44.25 Electronic ISBN: 978-1-4704-6502-5 Product Code: DOL/57.E List Price:$59.00
MAA Member Price: $44.25 AMS Member Price:$44.25
Bundle Print and Electronic Formats and Save!
This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.
List Price: $88.50 MAA Member Price:$66.38
AMS Member Price: $66.38 Click above image for expanded view A Gateway to Number Theory: Applying the Power of Algebraic Curves Keith Kendig Cleveland State University, Cleveland, OH MAA Press: An Imprint of the American Mathematical Society Available Formats:  Softcover ISBN: 978-1-4704-5622-1 Product Code: DOL/57  List Price:$59.00 MAA Member Price: $44.25 AMS Member Price:$44.25
 Electronic ISBN: 978-1-4704-6502-5 Product Code: DOL/57.E
 List Price: $59.00 MAA Member Price:$44.25 AMS Member Price: $44.25 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$88.50 MAA Member Price: $66.38 AMS Member Price:$66.38
• Book Details

Dolciani Mathematical Expositions
Volume: 572021; 207 pp
MSC: Primary 11; 14;

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.

• Chapters
• A marriage for the ages
• Viewing the whole algebraic curve
• Entering the world of elliptic curves
• Every elliptic curve is a group!
• A million-dollar challenge
• Every real elliptic curve lives in a donut
• The genus
• In conclusion $\ldots$
• What is a smooth complex curve?
• Algebraic curves in the disk model
• Some code for this books’s programs

• Reviews

• This book solves homogeneous diophantine equations in three variables (e.g., a2 + b2 = 2c2). The technique is to set x = a/c and y = b/c to arrive at a polynomial p(x, y) = 0 and then investigate rational points on the corresponding curve. The main investigation is of equations of degree 3, which correspond to elliptic curves. This pleasant and accessible journey continues into curves over C and the topology of algebraic curves. There are plenty of concrete examples, plus code (in GeoGebra, Maple, and Mathematica) for creating animations and solving the equations.

Mathematics Magazine
• Requests

Review Copy – for reviewers who would like to review an AMS book
Accessibility – to request an alternate format of an AMS title
Volume: 572021; 207 pp
MSC: Primary 11; 14;

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.

• Chapters
• A marriage for the ages
• Viewing the whole algebraic curve
• Entering the world of elliptic curves
• Every elliptic curve is a group!
• A million-dollar challenge
• Every real elliptic curve lives in a donut
• The genus
• In conclusion $\ldots$
• What is a smooth complex curve?
• Algebraic curves in the disk model
• Some code for this books’s programs
• This book solves homogeneous diophantine equations in three variables (e.g., a2 + b2 = 2c2). The technique is to set x = a/c and y = b/c to arrive at a polynomial p(x, y) = 0 and then investigate rational points on the corresponding curve. The main investigation is of equations of degree 3, which correspond to elliptic curves. This pleasant and accessible journey continues into curves over C and the topology of algebraic curves. There are plenty of concrete examples, plus code (in GeoGebra, Maple, and Mathematica) for creating animations and solving the equations.

Mathematics Magazine
Review Copy – for reviewers who would like to review an AMS book
Accessibility – to request an alternate format of an AMS title
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