ics141-lecture16-IntegersAlgorithms

Ics141-lecture16-Int - 16-1ICS 141 Discrete Mathematics I(Spr 2008)University of HawaiiICS141 Discrete Mathematics for Computer Science IDepartment

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Unformatted text preview: 16-1ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiICS141: Discrete Mathematics for Computer Science IDepartment of Information and Computer SciencesUniversity of HawaiiStephen Y. Itoga16-2ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiLecture 16Chapter 3. The Fundamentals3.5 Primes and Greatest Common Divisors 3.6 Integers and Algorithms16-3ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiSome 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. Baek16-4ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiReview: Prime NumbersAn integer p 1is primeiff the only positive factors of pare 1 and pitself.Some primes: 2,3,5,7,11,13,...Non-prime integers greater than 1 are called composite, because they can be composedby multiplying two integers greater than 1.16-5ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiPrime FactorizationExample 4: Fine the prime factorization of 7007 ( )Perform division of 7007 by successive primes7007 / 7 = 1001(7007 = 7·1001)Perform division of 1001 by successive primes beginning with 71001 / 7 = 143(7007 = 7·7·143)Perform division of 143 by successive primes beginning with 7143 / 11 = 13(7007 = 7·7·11·13 = 72·11·13)83.77007≈16-6ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiPrime Numbers: TheoremsTheorem 3: There are infinitely many primes.Assume that there are only finitely many primes, p1, p2,…, pnLet Q = p1p2···pn+ 1Then, 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 pidivides Q (if pi|Q then pi|(Q – p1p2···pn), i.e. pi|1)Hence there is a prime not in the list p1, p2,…, pn, which is either Q itself or a prime factor of Q (CONTRADICTION!!)16-7ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiMersenne PrimesDefinition: A Mersenne primeis a prime number of the form 2p– 1, where pis prime....
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This note was uploaded on 09/12/2008 for the course ICS 141 taught by Professor Idk during the Fall '08 term at Hawaii.

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Ics141-lecture16-Int - 16-1ICS 141 Discrete Mathematics I(Spr 2008)University of HawaiiICS141 Discrete Mathematics for Computer Science IDepartment

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