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

French mathematician Pierre de Fermat grew to become most excellent 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 e-book was once written in honor of the four-hundredth anniversary of his beginning and is predicated on a chain of lectures given by way of the authors. the aim of this ebook is to supply readers with an summary of the numerous houses of Fermat numbers and to illustrate their a variety of appearances and purposes in parts equivalent to quantity conception, likelihood idea, geometry, and sign processing. This booklet introduces a basic mathematical viewers to uncomplicated mathematical rules and algebraic equipment attached with the Fermat numbers and should offer worthy analyzing 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 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 collage of the United States in Washington, D. C.

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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). 4) is uniquely determined modulo M. 8. 3), where gcd(3,5, 7) = 1, we find the solution X = 23, which is the smallest in positive integers. 5 illustrates the cube 23 x 23 x 23, which is decomposed into blocks. Most of them are 3 x 5 x 7 blocks associated with the moduli 3, 5, and 7. There is one 2 x 3 x 2 block associated with the remainders 2, 3, and 2, and other mixed blocks associated with both moduli and remainders.

5. Primality of Fermat Numbers I have found that numbers of the form 22 = + 1 are always prime numbers and have long since signified to analysts the truth of this theorem . Pierre de Fermat in his letter to Father Marin Mersenne on December 25, 1640, [Mahoney, p. 140J. 1. Notice that the number 223 + 1 = 28 + 1 is prime, but the numbers 23 + 1 and 228 + 1 are composite (cf. Appendix A). This example shows that if 2n + 1 is prime, then 2 2n + 1 need not be prime and vice versa (see [Sierpinski, 1970, Problem 141]).

13. 11) = ordda, then 18 17 lectures on Fermat numbers for n = ke, k E {I, 2, ... 11) holds only for these exponents. Proof. If n = ke, an _ 1 then = a ke _ 1 = (a e _ 1) ( a e(k -1) + ... 11) is valid due to the previous definition. Assume, for an instant, that d I a ke + h -1 for some k E {I, 2, ... } and 0 Then (a ke + h _ 1) - (a ke - 1) = ake(a h - 1). < h < e. 11), gcd(d, a ke ) = 1, we get that a h -1 is divisible by d. This contradicts the minimality of e. 0 Let p be a prime. By Fermat's little theorem, the maximum order modulo p of any integer a coprime to p is p - 1.