Title: Gateway to Number Theory
Description: "Challenge: Can you find all the integers a, b, c satisfying 2a℗ø+3b℗ø=5c℗ø? 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℗ø+3b℗ø=5c℗ø. 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℗ø+3b℗ø=5c℗ø 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."--Publisher description.
Subjects: Mathematics, Diophantine equations, Algebraic Curves, Number theory, Number theory -- Instructional exposition (textbooks, tutorial papers, etc.)., Number theory -- Diophantine equations [See also 11Gxx, 14Gxx] -- Linear equations, Number theory -- Diophantine equations [See also 11Gxx, 14Gxx] -- Quadratic and bilinear equations, Number theory -- Diophantine equations [See also 11Gxx, 14Gxx] -- Cubic and quartic equations, Number theory -- Arithmetic algebraic geometry (Diophantine geometry) [See also 11Dxx, 14Gxx, 14Kxx] -- Elliptic curves over global fields [See also 14H52], Algebraic geometry -- Curves -- Special curves and curves of low genus, Algebraic geometry -- Curves -- Elliptic curves [See also 11G05, 11G07, 14Kxx]