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Unformatted text preview: Introductory Number Theory. Homework #2 All numbers are integers. 1. The Fibonacci sequence { F n } is defined recursively as follows: F 1 = 1, F 2 = 1, and then F n = F n − 1 + F n − 2 if n ≥ 3. Prove: gcd( F n , F n − 1 ) = 1for all n ∈ N . Use induction! 2. Prime power factorization. One way of emphasizing the uniqueness of the prime factorization of a number n ≥ 0 is to order the primes by size, group equals together as powers. One obtains the following theorem. Theorem. (Prime power factorization). Every natural number n ≥ 2 can be written uniquely in the form n = p e 1 1 ··· p e r r , where p 1 , . . . , p r are primes such that 2 ≤ p 1 < ··· < p r and e 1 , . . . , e r are positive integers. Uniqueness here means that if one also has n = q f 1 1 ··· q f s s , where q 1 , . . . , q s are primes such that 2 ≤ q 1 < ··· < q s and f 1 , . . . , f s are positive integers, then r = s ; p i = q i for i = 1 , . . . , r ; e i = f i for i = 1 , . . . , r....
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 Summer '11
 STAFF
 Prime number, lowest terms, prime power factorization

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