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17 lectures on Fermat numbers. From number theory to geometry. With a foreword by Alena Šolcová. (English) Zbl 1010.11002
CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. 9. New York, NY: Springer. xxiv, 257 p. (2001).
This is a nearly encyclopaedic work on Fermat numbers (natural numbers, especially primes of the form \(p=2^n+1)\). Fermat discovered that if \(p\) is a prime number, \(n\) must be a power of 2. But not every \(p\) of the form \(p=F(m)= 2^{2^m}+1\) is a prime. The first counterexample was given by Euler for \(m=5\).
After this curiosity Gauss found the connection to geometry with the construction of regular polytopes by ruler and compass. All this is explained in detail. So the influence of number theory on geometry begins to evolve.
Of course, Fermat’s “little” theorem has its prominent place. By the way of the proof some peculiar lemmata follow. In the chapter “Primality of Fermat numbers”, perhaps the heart of the book, we find the proof of two of the authors [M. Křižek and L. Somer, Math. Bohem. 126, 541-549 (2001; Zbl 0993.11002)] that for an odd \(n\geq 3\), \(F(n)\) is a Fermat prime if and only if the set of primitive roots modulo \(n\) is equal to the set of quadratic nonresidues modulo \(n\).
In the chapter “Fermat number transform and other applications” the authors introduce the use of Fermat numbers in number-theoretic transforms, in binary arithmetic modulo \(F(m)\) (which leads to fast multiplication of large numbers), in pseudorandom number generators, in minimal perfect hashing schemes, in the chiral Potts model, and in an analysis of the logistic equation by means of divisors of Fermat numbers – leaving us at the evolution of chaos.
The book begins with an essay on Fermat’s life and work. The new discussion about his year of birth [most probably 1507 instead of 1502, see K. Barner, Mitt. Dtsch. Math.-Ver. 2001, No. 3, 12-26 (2001)] is not included. The solution of Fermat’s last problem by A. Wiles is mentioned (more would not fit into the scope of this book). The table of contents placed after the foreword (on p. xxiii) is hard to find and, sorry to say, with wrong page numbers.
But there are interesting other tables, e.g. of Fermat numbers and their prime factors, and an appendix on Mersenne numbers (primes of the shape \(2^n-1)\) and their relations to even perfect numbers.
The book concludes with some pictures as “Reminiscence of Pierre de Fermat”, an extensive bibliography, and some useful website sources.

MSC:
11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
11A51 Factorization; primality
11A07 Congruences; primitive roots; residue systems
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