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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 10/23/2010 for the course MATH math 115A taught by Professor Anne during the Spring '10 term at UC Davis.

Page1 / 3

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online