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homework2

# homework2 - integers greater than 1 7 Use the Euclidean and...

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THEORY OF NUMBERS, Math 115 A Homework 2 Due Wednesday October 12 1. Use the sieve of Eratosthenes to find all primes less than 250 2. Find all primes that are difference of the fourth powers of two integers. 3. Let Q n = p 1 p 2 ...p n + 1 where p 1 , . . . , p n are the n smallest primes. Deter- mine the biggest prime factor of Q n for n 6. 4. Find the greatest common divisor of each of the following sets of integers: (a) 100, 121, (b) 1001, 289 (c) 6,15,21. For each two items please write the gcd as a linear combination of the numbers. 5. Show that two consecutive Fibonacci numbers f n and f n +1 are always relatively prime. Show that the gcd of f n and f n +3 is at most 2. 6. Every positive integer greater than 6 is the sum of two relatively prime
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Unformatted text preview: integers greater than 1. 7. Use the Euclidean and extended Euclidean algorithms to ﬁnd the gcd of the following pairs of numbers and to represent the gcd as a linear combination of the input data: a) (45 , 75), b) (666 , 1414), c) (102 , 222) d) (20785 , 44350). 8. Let m, n be positive integers and let a be an integer greater than 1. Show that ( a m-1 , a n-1) = a ( m,n )-1. 9. Show that every positive integer can be written as the product of possibly a square and a square-free integer. Recall a square free integer is an integer that is not divisible by any perfect square other than 1. 1...
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