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

French mathematician Pierre de Fermat turned prime for his pioneering paintings within the zone of quantity thought. 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 delivery and is predicated on a chain of lectures given by way of the authors. the aim of this e-book is to supply readers with an outline of the numerous homes of Fermat numbers and to illustrate their a variety of appearances and functions in parts reminiscent of quantity conception, likelihood conception, geometry, and sign processing. This publication introduces a basic mathematical viewers to simple mathematical principles and algebraic equipment attached with the Fermat numbers and should supply worthy studying for the novice 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 United States in Washington, D. C.

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**Extra resources for 17 Lectures on Fermat Numbers: From Number Theory to Geometry**

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All these examples can be generalized as follows: 2. 7 (Chinese Remainder Theorem). Let ml, m2, ... , mk be pairwise coprime natural numbers. 4) where the where Ti'S T2 (mod m2), are integers, there exists one and only one solution x modulo M, Proof. First we prove the existence of a solution x. 5) Since mi and Mi are coprime, there exist integers Yi, i = 1,2, ... 6) MiYi == 1 (mod m;). 4), we choose i E {l, ... , k}. 6) by Ti. 4). Then Xl == X2 (mod m;) for each i = 1, ... , k. Since mi are pairwise coprime, we have Xl == X2 (mod M).

If 1 < m < n, then m is said to be a nontrivial divisor of n. We say that m j exactly divides n, and write m j lin, if m j I n, but m H1 tn. Further, we introduce the notion of congruence, which was invented by C. F. Gauss. , in cryptography (the famous RSA method; see [Rivest, Shamir, Adleman]) . 4, and Chapter 15) . Let a, b, m be given integers and m 2 1. , a - b is divisible by m (d. 4), we write a == b (mod m) and say that a is congruent to b modulo mj b is called a residue of a modulo m. It is clear that there are exactly m distinct incongruent residues modulo m .

16. According to a well-known theorem of Sophie Germain, every number of the form a 4 + 4 is composite if a > 1. Indeed, Setting a = 22 =-2 for m ::=: 2, we find that a 2 + 1 = F m - 1 , a 4 + 1 therefore, Fm + 3 = (Frn - 1 + 2Fm- 2 - 1) (Fm- 1 - 2Fm- 2 + 3), where both factors are even and greater than 2 for m > 2. 17. Every Fermat number F m in the binary system has the form 1000 ... 0001 with 2m - 1 zeros inside (cf. [Leyendekkers, Shannon]). 18. , 1973, 1989], [Wei I], [Williams, 1998]. 4. The Most Beautiful Theorems on Fermat Numbers Whenever there is a number, there is beauty.