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

French mathematician Pierre de Fermat turned prime 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 e-book was once written in honor of the four-hundredth anniversary of his delivery and relies on a sequence of lectures given through the authors. the aim of this ebook is to supply readers with an summary of the numerous homes of Fermat numbers and to illustrate their a variety of appearances and functions in parts similar to quantity concept, likelihood idea, geometry, and sign processing. This e-book introduces a normal mathematical viewers to uncomplicated mathematical principles and algebraic equipment attached with the Fermat numbers and should supply valuable 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 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 US in Washington, D. C.

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2) 3, 5, 15, 17, 51, 85, 255, 257, .... 1. 1. The regular triangle, pentagon, heptadecagon, 257-gon and 65537-gon. 5. The regular polygon with n ::::: 3 sides can be constructed with ruler and compass if and only if there exists an integer q ::::: 1 such that Proof. Suppose that the regular polygon with n ::::: 3 sides has a Euclidean construction. 3, where i::::: 0, j::::: 0, F m " is prime for s = I, ... 14), where a = q = a + 2m , °+if ml < ° i = and a = i - I if i ::::: 1. Then clearly, ¢( n) 2 m2 + ...

J, and 4. The most beautiful theorems on Fermat numbers 35 where a = 0 if i = 0 and a = i - I if i ~ 1. Let us suppose that ks ~ 2 for some S E {I, ... ,j}. 3), Ps I ¢(n), which contradicts the assumption that ¢(n) = 2Q • Thus, ks = 1 for S = 1, 2, ... , j. 3) that Ps - 1 = 2ns , where ns ~ 1 for s = 1, ... , j. We claim that ns = 2m , for some ms ~ O. Suppose to the contrary that there exists s E {I, ... ,j} such that ns = k£ for some odd prime £ and some positive integer k. 3), 2 k € + 1 is composite which contradicts the fact that Ps is prime.

11 . 17 . 41 = 490688. \ (2 6 ), A(l1), A(17), A( 41)) = lcm(16, 10, 16,40) = 80, ¢( n) = ¢(2 6 )¢(11 )¢(17)¢( 41) = 32· 10 . 16·40 = 204800. The following theorem generalizes Euler's theorem. \( n) is a universal order modulo n. 22 (Carmichael). Let a , n EN. 18) if and aA(n) only if gcd( a, n) = 1. 19) Proof. 18) does not hold. Conversely, let gcd(a, n) = 1. 18) is clearly true when n = 1. Assume that n ~ 2. 20) ordna I A(n). 21 ) n = I1p7' , i=l 22 17 lectures on Fermat numbers where PI < P2 < ...