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intsu011h2s - Introductory Number Theory Homework#2 Some...

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Unformatted text preview: Introductory Number Theory. Homework #2 Some solutions. 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. I am not asking you to prove this, it is an immediate consequence of the Fundamental Theorem of Arithmetic; a different way of stating it. But I want you to be aware of it, understand it, because I plan to use it. Here is what I want you to prove: Prove that a number n ≥ 2 is a square if and only if in its prime power factorization n = p e 1 1 ··· p e r r all the exponents e 1 , . . . , e r are even. Proof. Assume first n is a perfect square, say n = m 2 . By the fundamental theorem of arithmetic, power version, we can write m = p f 1 1 ··· p f r r where p − 1 , . . . , p r are distinct primes and f 1 , . . . , f r natural numbers. Then n = m 2 = p 2 f 1 1 ··· p 2 f r r and, by the uniqueness of the prime power decomposition, this expression of n as product of powers of distinct primes is the prime power decomposition of n . Obviously, al the exponents 2 f 1 , . . . , 2 f r are even. Conversely, assume n = p e 1 1 ··· p e r r where all the p i ’s are distinct primes and all the e ′ i s are even. Writing e i = 2 f i for i = 1 , . . . , r ,, it is clear that n = ( p f 1 1 ··· p f r r ) 2 , hence is a square. 3. A rational number is one that can be written as a quotient of two integers. If r = a/b is a rational number b ̸ = 0, we say it is written in lowest terms or in reduced form iff b > 0 and gcd ( a, b ) = 1. It is easy to see that every rational number other then 0 has a unique representation in lowest terms. We will define a function f : Q → N as follows. If r > 0, and its representation in lowest terms is a/b , then both a, b > 0 and have no common factor. In this case we define f ( r ) = 2 a 3 b . If r < 0, then in lowest terms r = a/b with b > , a < 0....
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intsu011h2s - Introductory Number Theory Homework#2 Some...

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