Section 41 Primes, Factorization, and the Euclidean Algorithm第41节的素数，分解，与欧几里德算法.pptVIP

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Section 41 Primes, Factorization, and the Euclidean Algorithm第41节的素数，分解，与欧几里德算法.ppt

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Section 41 Primes, Factorization, and the Euclidean Algorithm第41节的素数，分解，与欧几里德算法

Section 4.1: Primes, Factorization, and the Euclidean Algorithm Practice HW (not to hand in) From Barr Text p. 160 # 6, 7, 8, 11, 12, 13 The purpose of the next two sections that we cover is to provide the mathematics background needed to understand the RSA Cryptosystem, which is a modern cryptosystem in wide use today. We start out by reviewing and expanding our study of prime numbers. Prime Numbers Recall that a prime number p is a number whose only divisors are 1 and itself (1 and p). A number that is not prime is said to be composite. The following set represents the set of primes that are less than 100: {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97…} A larger list of primes can be found in the Barr text on pp. 370-372. Facts About Primes 1. There are an infinite number of primes. Every natural number can be factored into a product of primes (Fundamental Theorem of Arithmetic). Determining the Primality of Larger Positive Integers Because of its use in cryptology and other applications, mathematical techniques for determining whether large numbers are prime have been targets of intense research. We study some elementary factors for determining the primality of numbers. Fact 2 is the only even prime. Any even number larger than 2 is not prime since 2 is a divisor. Example 1: Isprime? Solution: How do we determine if large positive integers are prime? The next example illustrates an elementary method for doing this? How do we determine if large positive integers are prime? The next example illustrates an elementary method for doing this? Example 2: Is 127 prime? Solution: Example 2 provides the justification for the following primality test for prime numbers. Square Root Test for Determining Prime Numbers Let n 1 be a natural number. If no prime number {2, 3, 5, 7, 11, 13, …} less than is a divisor of n, than n is prime. Example 3: Determine if 839 is prime. Solution: Example