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sol2 115A

# sol2 115A - MAT 115A University of California Fall 2010...

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MAT 115A University of California Fall 2010 Solutions Homework 2 1. Rosen 3.1 #4, pg. 74 Use the sieve of Eratosthenes or Sage to find all primes less than 200. Solution: The primes less than 200 are 2, 3, 5, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199. 2. Rosen 3.1 #5, pg. 74 Find all primes that are the difference of the fourth powers of two integers. Solution: Let n = x 4 - y 4 for x, y two integers where x > y . Then n = ( x - y )( x + y )( x 2 + y 2 ) and is divisible by x + y which cannot be 1 or n . Hence n cannot be a prime. So, there are no prime that is the difference of the fourth powers of two integers. 3. Rosen 3.1 #11, pg. 74 Let Q n = p 1 p 2 · · · p n +1, where p 1 , p 2 , . . . , p n are the n smallest primes. Determine the smallest prime factor of Q n for n 6. Do you think that Q n is prime infinitely often? ( Note: This is an unresolved question.) Solution: Q 1 = 3 , Q 2 = 7 , Q 3 = 31 , Q 4 = 211 , Q 5 = 2311 , Q 6 = 30031. The smallest prime factors are 3,7,31,211,2311, and 59, respectively. 4. Rosen 3.2 #3, pg. 86 Show that there are no “prime triplets”, that is, primes p , p + 2, and p + 4, other than 3,5, and 7. Solution: Suppose p, p + 2, and p + 4 were all prime. By divison al- gorithm we know p = 3 k or 3 k + 1 or 3 k + 2 for some integer

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