Gateway to Number Theory: Applying the Power of Algebraic Curves

The challenge is to find all integers $a$, $b$, and $c$ satisfying $2a^{2} + 3b^{2} = 5c^{2}$. This is done by transforming a degree-two equation into a plane curve defined by a polynomial, which makes obtaining solutions far easier. The book also addresses degree-three equations, which translate into elliptic curves, and includes many pictures and examples to make the material meaningful and easy to grasp. It is an ideal book for an introductory course on number theory and can also be self-studied.

Format: Paperback / softback
Length: 207 pages
Publication date: 30 June 2021
Publisher: American Mathematical Society

Rewritten text:

The challenge of finding all integers $a$, $b$, and $c$ satisfying the equation $2a^{2} + 3b^{2} = 5c^{2}$ is a fascinating and intricate problem in number theory. While it may appear straightforward at first glance, the sheer number of possibilities and the fact that each of $a$, $b$, and $c$ can be large make it a daunting task. However, a major breakthrough in number theory has made it possible to solve this problem with ease.

The key to this solution lies in transforming a degree-two equation in integer variables $a$, $b$, and $c$ into a plane curve defined by a polynomial. By working with the curve, solutions can be obtained more easily, and the geometric solutions can then be translated back into integers. This method simplifies complex problems and typically requires only a basic understanding of high school math.

In addition to equations of degree two, the book also addresses degree-three equations, a branch of number theory that has gained significant attention in recent years. These problems translate into elliptic curves, which are mathematical objects with fascinating properties. The book includes numerous pictures and illustrations to enhance the understanding of the material, making it accessible to students of all levels.

This book is an excellent resource for introductory courses on number theory, as well as for self-studying. The many solved examples, illustrations, and exercises make it an ideal choice for a capstone course, where students can apply their knowledge and skills to solve complex problems.

In conclusion, the challenge of finding all integers $a$, $b$, and $c$ satisfying $2a^{2} + 3b^{2} = 5c^{2}$ has been successfully overcome through the use of a novel approach based on plane. This book provides a comprehensive and accessible introduction to number theory, making it an invaluable tool for students and scholars alike.

Weight: 316g
Dimension: 153 x 232 x 15 (mm)
ISBN-13: 9781470456221