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# hw3 - allowed to be a prime number 4 Find the ±rst...

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MAT 312/AMS 351 – Fall 2010 Homework 3 1. Prove (by induction, or otherwise) that ( a - b ) | a n - b n for any integers a > b . A proof by induction would start with a 2 - b 2 = ( a - b )( a + b ), note that a 3 - b 3 = ( a - b )( a 2 + ab + b 2 ), use this form to conjecture the general statement and then prove it. 2. Use Exercise 1 to show that if 2 n - 1 is prime, then n must be prime. [The converse is not true in general, but a prime of the form 2 n - 1 is called a Mersenne prime . The largest prime known 2 43112609 - 1, discovered in 2008, is a Mersenne prime. Writing it out would require 12978189 decimal digits.] 3. What would happen to the Fundamental Theorem of Algebra if 1 were
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Unformatted text preview: allowed to be a prime number? 4. Find the ±rst positive integer value of n such that the formula n 2 + n +29 does not result in a prime number. 5. Consider an integer c with prime factorization c = p i 1 1 p i 2 2 · · · p i k k , with p 1 , . . . p k distinct. Show that in the prime factorization of c n , all the exponents are divisible by n . 6. Use the last execise to show that if an n-th power is the product of two relatively prime factors: c n = ab, ( a, b ) = 1 then each of the factors is itself an n-th power. 1...
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