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

ISBN-10: 0387218505

ISBN-13: 9780387218502

ISBN-10: 1441929525

ISBN-13: 9781441929525

French mathematician Pierre de Fermat grew to become ideal for his pioneering paintings within the region of quantity thought. His paintings with numbers has been attracting the eye of beginner mathematicians for over 350 years. This publication used to be written in honor of the four-hundredth anniversary of his delivery and is predicated on a sequence of lectures given by means of the authors. the aim of this e-book is to supply readers with an outline of the various houses of Fermat numbers and to illustrate their a number of appearances and functions in components akin to quantity idea, likelihood thought, geometry, and sign processing. This ebook introduces a normal mathematical viewers to uncomplicated mathematical principles and algebraic tools attached with the Fermat numbers and should supply helpful analyzing 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 United States in Washington, D. C.

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

Sample text

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. 3). 5 and [Martzloff, p. 317]). 6) can be transformed into that of finding the solution of the Diophantine equation MiYi - miVi = 1 for unknowns Yi and Vi.

It is easy to show that the converse assertion is not true. To see this, we take for instance a = 2 and n = 341. By inspection, 2 10 == 1 (mod 341), and thus 2340 == 1 (mod 341). Multiplying the last congruence by 2, we find that 341 I 2 341 _2. Hence, the converse of Fermat's little theorem (compare with [Robinson, 1957bJ) does not hold, since n = 341 = 31·11 is composite. There are only two more composite numbers with three digits, n = 561 and n = 645, for which n I 2 n - 2 (more details can be found in Chapter 12).

Any odd number can be written as the difference of two squares, 2n + 1 = (n + 1)2 - n 2 . Hence, any Fermat number can be written as the difference of two squares, namely Fm = 22 =+ 1 = (= 22 -1 +1 )2 _ (22 = )2 -1 . 6». 14. No Fermat prime can be expressed as tIle difference of two pth powers, where p is an odd prime. Proof . 9, > b. Since Fm is prime, we see that a - b = 1. By Fermat's Little Theorem Fm = aP - bP == a - b == 1 (mod p). Thus, which is impossible. Consequently, Fm cannot be represented as a difference of two pth powers.

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17 Lectures on Fermat Numbers: From Number Theory to Geometry by Michal Krizek, Florian Luca, Lawrence Somer, A. Solcova


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