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17 Lectures on Fermat Numbers: From Number Theory to Geometry

Michael Křižek, Florian Luca and Lawrence Somer
Springer Verlag
Publication Date: 
Number of Pages: 
CMS Books in Mathematics
[Reviewed by
Underwood Dudley
, on
Fermat numbers are Fn = 22n + 1, n = 0, 1, 2, … . Fermat incorrectly stated that they were all prime, but Euler showed that F5 was divisible by 641. From there up to F32 they are all composite and it is very likely that only finitely many are prime. Gauss showed that the number of sides of a regular polygon that could be constructed with straightedge and compass alone had to be a power of 2 times a product of distinct Fermat primes. Fermat numbers have many properties and even have applications.

This admirable book contains what must be everything that is worth knowing about Fermat numbers. (There are eighteen pages of references.) Though the authors claim it is for a general mathematical audience, readers should be familiar with elementary number theory. It contains a wealth of fascinating results, e.g., that 78557×2n + 1 is composite for all n, and that the rows of Pascal’s triangle modulo 2 give, when read as integers in base 2, the sequence of the number of sides of constructible regular polygons. The Foreword, by Alena Å olcová, is an excellent account of Fermat’s life and works.

The authors, and the series editors, Jonathan and Peter Borwein, deserve credit and praise for producing what will be the definitive work on the subject for many years to come.

After retiring from DePauw University, Woody Dudley is living in Florida and continues to be active in the MAA and in teaching calculus.

Foreword by Alena Solcov  * Table of Contents * Preface * Glossary of Symbols * Introduction * Fundamentals of Number Theory * Basic Properties of Fermat Numbers * The Most Beautiful Theorems on Fermat Numbers * Primality of Fermat Numbers * Divisibility of Fermat Numbers * Factors of Fermat Numbers * Connection With the Pascal Triangle * Miscellaneous Results * The Irrationality of the Sum of Some Reciprocals * Fermat Primes and a Diophantine Equation * Fermat's Little Theorem, Pseudoprimes, and Super-Pseudoprimes * Generalizations of Fermat Numbers * Open Problems * Fermat Number Transform and Other Applications * The Proof of Gauss's Theorem * Euclidean Constructions of the Regular Heptadecagon * Appendix A, B, C * References * Web Site Sources * Name Index * Subject Index