The mathematician Marin Mersenne (1588-1648) studied numbers of the form 2j – 1. In his honor, such numbers are known as Mersenne numbers, and any prime of the form 2j – 1 is called a Mersenne prime. The applet below takes a positive integer as input, and determines if the input is prime or composite. Here we test to see if the Mersenne number corresponding to j = 6 is a prime:
The applet below makes it easy to test several Mersenne numbers at once. You simply enter a value M, and the output consists of three columns. The first column shows the value of j, the second column indicates whether or not the jth Mersenne number is prime, and the third column shows the value of the jth Mersenne number. The values of j run from 1 to M.
Use the above applet to help you investigate Mersenne primes.
Research Question 1
Address the three questions given in the introduction of this chapter for Mersenne numbers:
1. Under what conditions is an integer of this form always prime?
2. Under what conditions is an integer of this form always composite?
3. Are there infinitely many primes of this form?6.2.1 An Application of Mersenne Primes
There is a connection between Mersenne primes and perfect numbers. A perfect number is an integer which is equal to the sum of its positive divisors less than itself. For example, if n = 6, then the positive divisors of n that are less than n are 1, 2, and 3. As
1 + 2 + 3 = 6, we see that 6 is perfect. We can test if a number is perfect with the following applet:
On the other hand, 7 is not perfect. Verify this by hand, and then execute the applet.
The applet below does tests each number from 1 to M, and reports those that are perfect:
In the next Research Question, you will be trying to find the connection between Mersenne primes and perfect numbers. In the course of your investigation, you may find it useful to factor integers. The applet below will do this automatically.
Research Question 2
How are perfect numbers related to Mersenne numbers?
Section 6.1 | Section 6.2 | Section 6.3 | Section 6.4
Copyright © 2001 by W. H. Freeman and Company