17 Lectures on Fermat Numbers: From Number Theory to Geometry

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Springer Science & Business Media, Mar 14, 2013 - Mathematics - 257 pages
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The pioneering work of French mathematician Pierre de Fermat has attracted the attention of mathematicians for over 350 years. This book was written in honor of the 400th anniversary of his birth, providing readers with an overview of the many properties of Fermat numbers and demonstrating their applications in areas such as number theory, probability theory, geometry, and signal processing. This book introduces a general mathematical audience to basic mathematical ideas and algebraic methods connected with the Fermat numbers.
 

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Contents

Introduction
1
Fundamentals of Number Theory
9
Basic Properties of Fermat Numbers
27
The Most Beautiful Theorems on Fermat Numbers
34
Primality of Fermat Numbers
41
Divisibility of Fermat Numbers
59
Factors of Fermat Numbers
71
Connection of Fermat Numbers with Pascals Triangle
81
Fermats Little Theorem Pseudoprimes and Superpseudoprimes
130
Generalizations of Fermat Numbers
149
Open Problems
158
Fermat Number Transform and Other Applications
167
The Proof of Gausss Theorem
189
Euclidean Construction of the Regular Heptadecagon
195
Appendix
209
B Mersenne Numbers
213

Miscellaneous Results
94
The Irrationality of the Sum of Some Reciprocals
104
Fermat Primes and a Diophantine Equation
119
Remembrance of Pierre de Fermat
218
References
226
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