Softcover ISBN:  9781470456221 
Product Code:  DOL/57 
List Price:  $59.00 
MAA Member Price:  $44.25 
AMS Member Price:  $44.25 
Electronic ISBN:  9781470465025 
Product Code:  DOL/57.E 
List Price:  $59.00 
MAA Member Price:  $44.25 
AMS Member Price:  $44.25 

Book DetailsDolciani Mathematical ExpositionsVolume: 57; 2021; 207 ppMSC: 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 degreetwo 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 degreethree 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 selfstudying the book an option for students, thus becoming a natural candidate for a capstone course.ReadershipUndergraduate and graduate students interested in elliptic curves.

Table of Contents

Chapters

A marriage for the ages

Viewing the whole algebraic curve

Entering the world of elliptic curves

Every elliptic curve is a group!

A milliondollar 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


Additional Material

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


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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 degreetwo 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 degreethree 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 selfstudying the book an option for students, thus becoming a natural candidate for a capstone course.
Undergraduate and graduate students interested in elliptic curves.

Chapters

A marriage for the ages

Viewing the whole algebraic curve

Entering the world of elliptic curves

Every elliptic curve is a group!

A milliondollar 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