Problems for Section 33 1 Find the next perfect number after

Problems for Section 3.3. 1. Find the next perfect number after 496. 2. Extend the table given in this section to run from k 7 to 10. 3. An early author stated that if k 1 is odd then 2 1 is prime. Show that this is false. 4. An early author stated that if k is prime then 2 1 is prime. Show that this is false. In the neat five problems (a) give the prime power factorization of 2 1; (b) list all numbers 2 1, where d divides k, 1 d k.

Solution

Q1) Using the mersenne prime number p and making sure that the number is of the form 2^(p-1) * (2^p -1)

The next perfect number will be 8128

Q6)

Number = 2^(6) - 1 = 63

Factors of 63 are 1,3,7,21

3 can be written in the form of (2^2-1), and 2 divides 6

10)

For k=1, Number = 2^(1) + 1 = 3, which is a prime number

For k=2, Number = 2^(2) + 1 = 5, which is a prime number

For k=3, Number = 2^(3) + 1 = 9, which is not a prime number

For k=4, Number = 2^(4) + 1 = 17, which is a prime number

For k=5.Number = 2^(5) + 1 = 33, which is not a prime number

For k=6, Number = 2^(6) + 1 = 65, which is not a prime number

For k=7, Number = 2^(7) + 1 = 129, which is not a prime number

For k=8, Number = 2^(8) + 1 = 257, which is a prime number

For k=9, Number = 2^(9) + 1 = 513, 513 is not a prime number

For k=10, Number = 2^(10) + 1 = 1025, which is not a prime number