If a Mersenne number turns out to be a prime number, then it is called a Mersenne prime.

Just to summarize what we have done so far, let's examine Table 2 again. This time, we will express the numbers in powers of two, and delete those rows that do not carry perfect numbers.

(a) Complete the following table, expressing the first five Mersenne primes and perfect numbers in powers of two.

Table 4:  The first five Mersenne primes and the corresponding perfect numbers.

(b) Two perfect numbers were discovered in 1588, both by Cataldi. These two perfect numbers can be obtained from the Mersenne primes M₁₇ = 2¹⁷ - 1 and M₁₉ = 2¹⁹ - 1. Can you compute these two perfect numbers with the help of your calculator? 

(c) Do you think M₂₁ is a Mersenne prime?

By now, you should have realized why numbers of the form 2ⁿ - 1 have so much appeal. Whenever a prime number of this form is found, a perfect number is immediately obtained, as was proven by Euclid.

So the search for perfect numbers became the search for more Mersenne primes, i.e. prime numbers of the form 2ⁿ - 1. But this turned out to be a very challenging task, since, as a number gets bigger it becomes extremely tedious to test for its primacy, without the use of computing devices, like the electronic calculators we have today.

Father Marin Mersenne (1588-1648), a French monk after whom Mersenne numbers are named, claimed that the numbers 2ⁿ - 1 were prime for

and were composite for all other positive integers n < 257. As you will see later, his claim was not entirely correct. Neither he nor his contemporaries were able to prove his claim.
 
 

Study carefully the list of Mersenne primes which have been established so far:

The list of integers which produce Mersenne primes are all prime numbers themselves. Naturally, we would immediately want to ascertain whether a number of the form 2ⁿ - 1 could be prime, if n is not prime. We will investigate this in the following exercise. 

(a) How many Mersenne primes were known during Mersenne's time?

(b) Without using calculator, show that M₁₄ = 2¹⁴ - 1 can be factorized. Do the same for M₂₆ = 2²⁶ - 1.
[Hint: Express M₁₄ as a difference of two squares.]

(c) Prove that, if n is an even number greater than 2, then 2ⁿ - 1 is composite.

*(d) Without using calculator, show that M₁₅ = 2¹⁵ - 1 can be factorized. Do the same for M₂₁ = 2²¹ - 1.

*(e) Prove that, if n is composite, then 2ⁿ - 1 is composite.

From the previous exercise, we learned that the only values of n which will produce Mersenne primes are the prime numbers themselves. So, to find Mersenne primes, we can totally disregard values of n which are not prime.

While it is necessary for n to be prime to ensure that 2ⁿ - 1 is prime, not every prime number value of n will produce a Mersenne prime. For example, the number M₁₁ is noticeably missing from the list:

So, the Mersenne primes seem to pick only on the prime numbers, and do so selectively. It decided to skip the next two, M₂₃ and M²⁹, before picking on M₃₁. As was mentioned previously, though Mersenne suggested 2³¹ - 1 was prime, he was not able to prove it.

The world waited another hundred years before the great Swiss mathematician, Leonard Euler (1707-1783), who, by the use of clever reasoning and trial division, showed that 2³¹ - 1 = 2147483647 is prime. Consequently, the discovery of the 8th perfect number, was attributed to Euler.

In 1811 Peter Barlow, commented on the perfect number 2³⁰(2³¹ - 1) that it "is the greatest that will be discovered, for as they are merely curious, without being useful, it is not likely that any person will attempt to find one beyond it." Obviously, no one in the late 1800's had any idea of the power of modern computers. What might we know about the machines of 50 years from now? 

(a) M₂₃ = 2²³ - 1 is not a prime. Verify this with the help of your calculator.

(b) M₂₉ = 2²⁹ - 1 is also not a prime. It can be quite hard to verify this, even with the use of a calculator. But you may give it a try.

(c) Test your eyesight using "The Perfect Eye Chart".

 
 

How were the larger perfect numbers found?

Most of the larger perfect numbers discovered after Euler were found in the computer age, with the help of electronic computing devices. By 1985, a total of 31 perfect numbers were known. By the fall of 2008, the 45th and 46th perfect number were discovered. You may even participate in finding the next perfect number by joining the GIMPS (Great Internet Mersenne Prime Search).

All the known perfect numbers are in the form 2^n-1(2ⁿ - 1), firmly established since Euclid's time. It was Euler who showed that there are no other even perfect numbers, except in the form of Euclid's formula.

Are there any odd perfect numbers? Nobody has found any, and nobody has been able to prove that there is none. All we know about the odd ones is that they must have at least 300 decimal digits and many factors. There probably aren't any!