Hilbert’s Tenth Problem

Hilbert’s tenth problem is one of 23 problems proposed by David Hilbert in 1900 at the International Congress of Mathematicians in Paris. These problems gave focus for the exponential development of mathematical thought over the following century. The tenth problem asked for a general algorithm to determine if a given Diophantine equation has a solution in integers. It was finally resolved in a series of papers written by Julia Robinson, Martin Davis, Hilary Putnam, and finally Yuri Matiyasevich in 1970. They showed that no such algorithm exists. This book is an exposition of this remarkable achievement. Often, the solution to a famous problem involves formidable background. Surprisingly, the solution of Hilbert’s tenth problem does not. What is needed is only some elementary number theory and rudimentary logic. In this book, the authors present the complete proof along with the romantic history that goes with it. Along the way, the reader is introduced to Cantor’s transfinite numbers, axiomatic set theory, Turing machines, and Gödel’s incompleteness theorems. Copious exercises are included at the end of each chapter to guide the student gently on this ascent. For the advanced student, the final chapter highlights recent developments and suggests future directions. The book is suitable for undergraduates and graduate students. It is essentially self-contained.

Table of Contents

Chapter 1. Cantor and Infinity

Chapter 2. Axiomatic Set Theory

Chapter 3. Elementary Number Theory

Chapter 4. Computability and Provability

Chapter 5. Hilbert’s Tenth Problem

Chapter 6. Applications of Hilbert’s Tenth Problem

Chapter 7. Hilbert’s Tenth Problem over Number Fields

Appendix A. Background Material

About the Author

M. Ram Murty: Queen s University, Kingston, ON, Canada, Brandon Fodden: Carleton University, Ottawa, ON, Canada