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

French mathematician Pierre de Fermat grew to become finest for his pioneering paintings within the sector of quantity idea. His paintings with numbers has been attracting the eye of novice mathematicians for over 350 years. This ebook used to be written in honor of the four-hundredth anniversary of his delivery and relies on a chain of lectures given through the authors. the aim of this publication is to supply readers with an summary of the numerous houses of Fermat numbers and to illustrate their a number of appearances and functions in parts equivalent to quantity concept, likelihood conception, geometry, and sign processing. This booklet introduces a normal mathematical viewers to uncomplicated mathematical rules and algebraic tools hooked up with the Fermat numbers and should supply necessary analyzing 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 college 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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**Example text**

9), which produces the remainder O. We obtain the same remainders (up to 2. Fundamentals of number theory 17 order) when dividing the sequence 1, ... ,p - 1 by p. 2), we arrive at aP-1(p - I)! == (p - I)! (mod pl. Consequently, (a P - 1 - 1)(p - I)! == 0 (mod p), and the first term on the left-hand side is thus divisible by the prime p, since pf(p-l)!. 10. 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.

M. 3 (Selfridge's Test). Let N > 1 and let the prime-power factorization of N - 1 be given by r IIp7 N - 1= i • i= l Then N is prime if and only if for each prime Pi, i E {1, ... , r }, there exists an integer ai > 1 such that (i) a["-l == 1 (mod N), .. ) cd 1 ( d N) ( II a (N-l)/p, 'F mo". i Proof. If N is prime, then there exists a primitive root a that satisfies conditions (i) and (ii) for i = 1, ... , r. Now assume that both conditions (i) and (ii) hold for i = 1, ... , r. It suffices to show that ¢(N) = N - 1.

7) p = k2 rn + 2 + 1, where k is a natural number. For a proof see the beginning of Chapter 6. 3]). 19. Ifm> 1, then any divisor d > 1 of a Fermat number Fm is of the form k2 rn + 2 + 1, where k is a natural number. Proof. 18, since any prime divisor p of d is congruent to 1 (mod 2m + 2 ), and the product of numbers each congruent to 1 (mod 2m+2) is also congruent to 1 (mod 2m+2). 20. 18 can be illustrated by an example, which was treated by A. E. Western already in 1903. He was searching for a natural number k such that k2 20 + 1 I F 18 .