Mersenne numbers

So far, we know that every odd prime number of the form 2k - 1 is a factor of an even perfect number, and also that every even perfect number has exactly one prime factor, which must be of the form 2k - 1. The search for additional even perfect numbers, then, boils down to an examination of numbers of the form 2k - 1, to determine which are prime.

Definition: A number of the form 2k - 1 is called a Mersenne number and is denoted by Mk.

We can rephrase our statement about even perfect numbers as follows:

Mk is prime if and only if (2k-1)Mk is perfect

To determine whether or not Mk is prime, we can always resort to the definition of prime numbers. Given enough time, Mk can be tested for each specific value of k to determine if it is prime. But you get the feeling pretty quickly that there really ought to be a better way. Here's what we know so far:

M8=255 is clearly composite, which suggests the possibility that the Mersenne numbers are alternately prime and composite, after an initial anomaly. Unfortunately, this guess is quickly disproved, since M9 = 511 = 7 . 73, which is composite.

A different conjecture, with no noticeable exceptions, is readily suggested by comparing the classification of Mk with the classification of k itself. So far, Mk is prime when k is 2, 3, 5 or 7, and Mk is composite when k is 4, 6, 8, or 9. Here's a conjecture that takes this into account:

Conjecture: Mk is prime if k is prime and is composite if k is composite.

We even have M1 = 1, the subscript matching the value, in the special case of the only number that is neither prime nor composite. This suggests that M10 should be composite, M11 should be prime, M12 composite, and so on. Note that there are two parts to the conjecture, so let's take them one at a time, the "composite" part first.

Here are some factorizations of composite Mersenne numbers:

kMkFactorization of Mk
4153 . 5
66332 . 7
82553 . 5 . 17
95117 . 73
1010233 . 11 . 31
Note that a composite Mersenne number always seems to have Mersenne primes among its factors. Specifically:

Again, there is a striking pattern among the Mersenne numbers and their subscripts, suggesting that:

if c is a factor of d, then Mc is a factor of Md

We'll prove this in class, which gives the proof of half of our conjecture, namely that if k is composite, then Mk is composite.

It was surprising to the first people who proved it that the other half of our conjecture turns out to be false! It is false for k=11 as well as for many (most) other prime values of k. Although 11 is prime, M11 is composite.

This is one of the classic examples in mathematics of a persuasive pattern turning out to be misleading. The incorrectness of such a persuasive conjecture, based on several verified examples, is a paramount example of why mathematics insists on rigorous proofs for every assertion, even the seemingly obvious ones. Simplicity of form, successful prediction of examples, and majority belief can all be wrong.

Where to we stand in our search for Mersenne prime numbers? We know we only have to consider Mersenne numbers with prime subscripts, and some additional narrowing can be effected by the form of the prime, it is basically the true that no pattern has yet been found and one must test all the Mersenne numbers with prime subscripts. Special algorithms for testing the primality of large numbers on computers have extended the list of known Mersenne primes, but no general characterization of Mersenne primes has yet been proved.

Finally, to round out our discussion of perfect numbers, consider the question of odd perfect numbers. Basically, nothing definitive is known, although computer searches up to around 100,000,000,000,000,000,000 haven't yielded any odd perfect numbers yet. Many mathematicians believe that no odd perfect numbers exist, but no proof of this statement exists yet.