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ics141-lecture16-IntegersAlgorithms

ics141-lecture16-IntegersAlgorithms - University of Hawaii...

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16-1 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii ICS141: Discrete Mathematics for Computer Science I Department of Information and Computer Sciences University of Hawaii Stephen Y. Itoga

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16-2 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii Lecture 16 Chapter 3. The Fundamentals 3.5 Primes and Greatest Common Divisors 3.6 Integers and Algorithms
16-3 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii Some material in these slides were taken/adapted from the slides made by Prof. Michael P. Frank and Prof. Jonathan L. Gross which are provided through the publisher of “Discrete Mathematics and Its Applications” written by Kenneth H. Rosen. Some slides were done by Prof. Baek

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16-4 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii Review: Prime Numbers An integer p 1 is prime iff the only positive factors of p are 1 and p itself. Some primes: 2,3,5,7,11,13,... Non-prime integers greater than 1 are called composite , because they can be composed by multiplying two integers greater than 1.
16-5 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii Prime Factorization Example 4 : Fine the prime factorization of 7007 ( ) Perform division of 7007 by successive primes 7007 / 7 = 1001 (7007 = 7·1001) Perform division of 1001 by successive primes beginning with 7 1001 / 7 = 143 (7007 = 7·7·143) Perform division of 143 by successive primes beginning with 7 143 / 11 = 13 (7007 = 7·7·11·13 = 7 2 ·11·13) 83.7 7007

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16-6 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii Prime Numbers: Theorems Theorem 3 : There are infinitely many primes. Assume that there are only finitely many primes, p 1 , p 2 ,…, p n Let Q = p 1 p 2 ···p n + 1 Then, Q is prime or it can be written as the product of two or more primes (by Fundamental Theorem of Arithmetic) None of the primes p i divides Q (if p i |Q then p i |(Q – p 1 p 2 ···p n ), i.e. p i |1) Hence there is a prime not in the list p 1 , p 2 ,…, p n , which is either Q itself or a prime factor of Q (CONTRADICTION!!)
16-7 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii Mersenne Primes Definition : A Mersenne prime is a prime number of the form 2 p – 1, where p is prime.

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