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

The pioneering paintings of Pierre de Fermat has attracted the eye of mathematicians for over 350 years. This booklet offers an outline of the various homes of Fermat numbers and demonstrates their purposes in components corresponding to quantity concept, likelihood concept, geometry, and sign processing. it truly is a terrific creation to the fundamental mathematical rules and algebraic tools hooked up with the Fermat numbers.

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Extra info for 17 Lectures on Fermat Numbers: From Number Theory to Geometry (CMS Books in Mathematics)

Example text

If p is a prime and ab 0 (mod p), then it is easy to sec that a 0 (mod p) or b = 0 (mod pl. We shall use this elementary fact in proving the next theorem, which is one of the most frequently used tools in number theory, as we shall see also in this book (sec the Subject Index). 9 (Fermat's Little Theorem). If a is a natural number and p a prime number, then p I aP - a. Proof. 9) 1 - - a = a(a P - 1 - 1). So let = l. l. Consider the finite sequence a, 2a, 3a, ... , (p - l)a, pa. 8). 9) yields p different remainders upon division by p.

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.

10) k :::: 3 and n :::: m + 2. 9) says that k and (i cannot be simultaneously small relative to Fm. 10]). Its assumptions are satisfied for a majority (about 85%) of all known prime factors of the Fermat numb ers and for all 151 known prime factors of Fm for 29::; m::; 382447 as of the beginning of 2000. 22 (Suyama). Let p = k2n + 1 divide Fm and let k2 n -(m+2j < 9 . 2m + 2 + 6. Then p is a prime. Proof. Suppose, to the contrary, that p is a product of two nontrivial factors. 10), and thus k2n + 1 :::: (3· 2m+2 + 1)2.

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