MATH 319 Problem Set 6

1. Test the integer 2,152,302,898,747 to see if it is a strong pseudoprime to the bases 2, 3, 5, 7, and 11.

2. Problem 16 on page 248 gives a criterion for a pair of integers to be amicable. Using the maple function isprime which returns true for a prime and false for a composite find three pairs of amicable numbers different from those listed in problem 15 on the same page.

3. For each of the primes p from 23 to 59 test 2^p – 1 for primality. For those that are primes find the related perfect number. Factor those that are not primes.

4. The following simple Maple program can be used to find pseudoprimes:

> for k from 300 to 400 do

> k,modp(2&^(k-1),k),isprime(k);

> od;

Run the program to find any pseudoprimes to base 2 between 100 and 200. What in the output tells you that you have a pseudoprime? Then find all pseudoprimes to either base 2 or 3 that are less than 1000.

5. From the text :

page 117 #2c, 4a,c,d and page 290 #2