By Michal Krizek, Florian Luca, Lawrence Somer, A. Solcova

French mathematician Pierre de Fermat turned optimum for his pioneering paintings within the zone of quantity conception. His paintings with numbers has been attracting the eye of novice mathematicians for over 350 years. This ebook was once written in honor of the four-hundredth anniversary of his beginning and relies on a chain of lectures given through the authors. the aim of this publication is to supply readers with an outline of the various houses of Fermat numbers and to illustrate their a variety of appearances and functions in parts akin to quantity thought, likelihood idea, geometry, and sign processing. This e-book introduces a basic mathematical viewers to uncomplicated mathematical principles and algebraic equipment attached with the Fermat numbers and should supply worthwhile studying for the beginner alike.

Michal Krizek is a senior researcher on the Mathematical Institute of the Academy of Sciences of the Czech Republic and affiliate Professor within the division of arithmetic and Physics at Charles collage in Prague. Florian Luca is a researcher on the Mathematical Institute of the UNAM in Morelia, Mexico. Lawrence Somer is a Professor of arithmetic on the Catholic college of the USA in Washington, D. C.

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263]. The fundamental theorem of arithmetic thus says that every natural number n > 1 can be written uniquely as a product of primes ql < q2 < ... < qs: = qr ' q~2 ... 1) n i=1 M. , 17 Lectures on Fermat Numbers © Springer Science+Business Media New York 2002 10 where 17 lectures on Fermat numbers Q; 2: 1 for i E {I, ... , s}. 3. 1). , the famous expression of the Riemann (-function via the product over all primes P, L 00 ((z) = 1 nZ = n=l 1 II 1 _ p-z' p where the sum converges for an arbitrary complex number z such that Re(z) > 1.

There are several other proofs of Fermat's little theorem. A very interesting approach to this is given in [Gutfreund, Little]. It uses an analogy with physical particles based on symmetry properties of Ising spin configurations. For further discussion about Fermat's little theorem and also about the Fermat quotient (a P - 1 - 1)/p see [Lepka]. 11. 10) (mod pl. 10) can be used to prove that a given number is composite without knowing any of its factors. 10) does not hold, then p is not a prime.

Suppose now that n is not a power of 2 and that 2n + 1 is a pseudoprime to the base 2. 5) 4. The most beautiful theorems on Fermat numbers since gcd(2, 2 n + 1) = 37 1. Note that and that 2n < 2n + 1. Let e = ord2n+12. Then e 2: n 2t == 1 (mod 2n + 1), then e I t. 13, if + 1), it follows that e = 2n. 13, we have 2n I 2n. However, this is impossible, since 2n is not a power of 2. 11. According to T. 4) and believed that this relation implies that all Fm are prime. According to [Kiss, p. 72], Janos Bolyai (1802-1860), one of the founders of nonEuclidean geometry, was the first who showed that F5 is a pseudoprime to the base 2.