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.
Author(s): Michal Krizek; Florian Luca; Lawrence Somer
Edition: Paperback
Publisher: Springer
Year: 2011
Language: English
Pages: 257
Tags: Number Theory, 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
