4.3 - Primes and Greatest Common Divisors

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1 4.3 - Primes and Greatest Common Divisors Introduction We focus on properties of integers and prime factors Primes Definition 1 An integer p greater than 1 is called prime if the only positive factors of p are 1 and p. A positive integer that is greater than 1 and is not prime is called composite. So integers such as 2, 3, 5, 7, 11 are all prime, while 6, 8, 9, 10, and 12 are all composite. Note that 1 is neither prime nor composite. So an integer n is composite if and only if there exists an integer a such that a n and 1 < a < n. Theorem 1 - The Fundamental Theorem of Arithmetic Every prime integer greater than 1 can be written uniquely as a prime or as the product of two or more primes where the prime factors are written in order of nondecreasing size. So for instance, 2 = 2, 3 = 3, 6 = 2ÿ3, 8 = 2ÿ2ÿ2 = 2 3, 9 = 3ÿ3 = 3 2, and 12 = 2ÿ2ÿ3 = 2 2 ÿ3. Trial Division We often need to determine if a number is prime Theorem 2 If n is a composite integer, then n has a prime divisor less than or equal to n. ü Example Show that 53 is prime.

2 2 Lecture_04_03.nb The Sieve of Eratosthenes The Sieve of Eratosthenes is used when we want to find all prime numbers less than or equal to some given integer n. From the list, circle the first prime number 2, and then eliminate all of its multiples. Then, circle the next prime number, 3 and eliminate all of its multiples. Continue until you have eliminated all the multiples of the primes less than or equal to n. The remaining entries are all prime (except for 1). Theorem 3 There are infinitely many primes. ü Example Prove Theorem What is the largest prime? For the last 300+ years, the largest known prime has been an integer of the form 2 p - 1, where p is prime. Primes of this form are called Mersenne primes. Recently, a new largest prime was found, Note that not all primes are of this form (for instance = 63 = 3ÿ3ÿ7), nor are all numbers of the form 2 p - 1 prime (for instance, 2 11 = 2047 = 23ÿ89). Given a number x, how many primes are less than x? Theorem 4 - The Prime Number Theorem x The ratio of the number of primes not exceeding x and approaches 1 as x grows without bound. ln x In number theory, the function phxl gives the number of primes less than or equal to x. So Theorem 4 says phxl lim xø = 1. J x N ln x

3 Lecture_04_03.nb 3 n πhxl x ln x π HxL xêln x Conjectures and Open Problems About Primes Is there a function f HnL such that f HnL is prime for all positive integers? It can be shown that there is no polynomial such that f HnL is prime for all positive integers. The Goldbach Conjecture states that every odd integer greater than 5 is the sum of three prime numbers. The Twin Prime Conjecture state that there are infinitely many twin primes (twin primes are prime numbers whose difference is 2). Greatest Common Divisors and Least Common Multiples Definition 2 Let a and b be integers, not both zero. The largest integer d such that d a and d b is called the greatest common divisor of a and b. The greatest common divisor of a and b is denoted by gcdha, bl. So gcdh18, 24L = 6 and gcdh14, 15L = 1. Definition 3 The integers a and b are relatively prime if their greatest common divisor is 1. So 14 and 15 are relatively prime since gcdh14, 15L = 1. Definition 4 The integers a 1, a 2,..., a n are pairwise relatively prime if gcdia i, a j M = 1 whenever 1 i < j n. So 13, 14 and 15 are pairwise relatively prime since gcdh13, 14L = 1, gcdh13, 15L = 1, and gcdh14, 15L = 1. Definition 5 The least common multiple of the positive integers a and b is the smallest positive integer that is divisible by both a and b. The least common multiple of a and b is denoted by lcmha, bl. So lcmh18, 24L = 72 and lcmh14, 15L = 210.

4 4 Lecture_04_03.nb Suppose that the prime factorizations of two positive integers a and b are given by then a = p 1 a 1 p 2 a 2 p n a n and b = p 1 b 1 p 2 b 2 p n b n gcdha, bl = p 1 minha 1, b 1 L p 2 minha 2, b 2 L p n minha n, b n L and lcmha, bl = p 1 maxha 1, b 1 L p 2 maxha 2, b 2 L p n maxha n, b n L Theorem 5 Let a and b be positive integers. Then ab= gcdha, blÿlcmha, bl. Note that 18ÿ24 = 432 and that gcdh18, 24LÿlcmH18, 24L = 6ÿ72 = 432. The Euclidean Algorithm Right now we need to know the prime factorization of two numbers to determine their greatest common divisors, this is not very efficient. The Euclidean algorithm gives us a more efficient way of doing this. It relies on the following result. Lemma 1 Let a = bq+ r, where a, b, q, and r are integers. Then gcdha, bl = gcdhb, rl. Algorithm 1 - The Euclidean Algorithm procedure gcd(a, b: positive integers) x := a y := b while y 0 r := x mod y x := y y := r return x{gcdha, bl = x} If a b then the number of division required by this algorithm is OHlog bl. ü Example Use the Euclidean Algorithm to calculate the following: (a) gcdh108, 48L (b) gcdh2436, 13L

5 Lecture_04_03.nb 5 gcds as Linear Combinations We can express the greatest common divisor of two positive integers a and b as a linear combination of the two positive integers a and b. Theorem 6 - Bezout s Theorem If a and b are positive integers, then there exists integers s and t such that gcdha, bl = sa+ tb. Definition 6 If a and b are positive integers, then integers s and t such that gcdha, bl = sa+ tb are called Bezout coefficients of a and b. Also, the equation gcdha, bl = sa+ tb is called Bezout s Identity. ü Example For each problem, express the gcdha, bl as a linear combination a and b. (a) gcdh108, 48L (b) gcdh2436, 13L Lemma 2 If a, b, and c are positive integers such that gcdha, bl = 1 and a bc, then a c. For instance, we know that gcdh3, 5L = 1 and we can show that where 195 = 5ÿ3843, so Lemma 3 If p is prime and p a 1 a 2 a n, where each a i is an integer, then p a i for some i. Theorem 7 Let m be a positive integer and let a, b, and c be integers. If acª bchmod ml and gcdhc, ml = 1, then a ª b Hmod ml. For instance, we can show that 6 ª 62 Hmod 7L. Since 6 = 3ÿ2, 62 = 31ÿ2, and gcdh2, 7L = 1, then 3 ª 31 Hmod 7L.