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

French mathematician Pierre de Fermat turned most desirable for his pioneering paintings within the quarter of quantity concept. His paintings with numbers has been attracting the eye of beginner mathematicians for over 350 years. This publication was once written in honor of the four-hundredth anniversary of his beginning and relies on a chain of lectures given by way of the authors. the aim of this publication is to supply readers with an outline of the numerous houses of Fermat numbers and to illustrate their various appearances and functions in components reminiscent of quantity conception, chance thought, geometry, and sign processing. This publication introduces a basic mathematical viewers to uncomplicated mathematical principles and algebraic tools attached with the Fermat numbers and may supply priceless 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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**Example text**

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. His proof was based on the observation that 2232 == 1 (mod F 5 ), where F5 = 232 + 1 = 641 . 6700417. 12. Let q = pn be a power of an odd prime p, where n 2: 1. Then the Fermat number Fm is divisible by q if and only if Proof. Suppose that q I Fm. Then q obviously also divides the number 22=+1 _ 1 = (22= + 1) (22= - 1). 13 for a = 2 it follows that ord q 2 :::; 2m+!

Let n 2 2 and a be integers such that gcd(a, n) = 1. If the quadratic congruence x 2 == a (mod n) has a solution x, then a is called a quadratic residue modulo n. Otherwise, a is called a quadratic nonresidue modulo n. 25. Let p be an odd prime. Then the Legendre symbol (~) is defined to be 1 if a is a quadratic residue modulo p, -1 if a is a quadratic nonresidue modulo p, and 0 if pia. The quadratic character of a with respect to p is thus expressed via the Legendre repeats periodically. For instance, symbol.

In Chapter 5 we will explore this phenomenon further. The theorem below appears in [Racli§]. 11 (Racli§). Let Fm be a prime and a an integer. Then every prime Fermat number less than or equal to Fm divides a F = - a. Fk Proof. Let be any prime Fermat number for which 0 ::; k ::; m. First suppose that I a. 14) Fk t a. By Fermat's little theorem, 31 3. 15), The following theorem is mentioned in [Kiss]. 12 (Bolyai). Every Fermat number Fm for m 6n - 1. Proof. We have to show that 6 I Fm Fm + 1. 4), + 1 = FoFl ...