17 Lectures on Fermat Numbers: From Number Theory to Geometry

Description

The pioneering work of Pierre de Fermat has attracted the attention of mathematicians for over 350 years. This book provides an overview of the many properties of Fermat numbers and demonstrates their applications in areas such as number theory, probability theory, geometry, and signal processing. It is an ideal introduction to the basic mathematical ideas and algebraic methods connected with the Fermat numbers. Contents Foreword vii Preface xix Glossary of Symbols xxi 1 Introduction 1 2 Fundamentals of Number Theory 9 3 Basic Properties of Fermat Numbers 27 4 The Most Beautiful Theorems on Fermat Numbers 34 5 Primality of Fermat Numbers 42 6 Divisibility of Fermat Numbers 60 7 Factors of Fermat Numbers 71 8 Connection of Fermat Numbers with Pascal's Triangle 81 9 Miscellaneous Results 96 10 The Irrationality of the Sum of Some Reciprocals 106 11 Fermat Primes and a Diophantine Equation 119 12 Fermat's Little Theorem, Pseudoprimes, and Superpseudoprimes 132 13 Generalizations of Fermat Numbers 149 14 Open Problems 160 15 Fermat Number Transform and Other Applications 167 16 The Proof of Gauss's Theorem 189 17 Euclidean Construction of the Regular Heptadecagon 195 Appendix 209 A Tables of Fermat Numbers and Their Prime Factors 209 B Mersenne Numbers 215 C Remembrance of Pierre de Fermat 220 References 228 Web Site Sources 243 Name Index 244 Subject Index 249