**Definition:** A number of the form 2^{k} - 1 is called a *
Mersenne number* and is denoted by M_{k}.

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

** M _{k} is prime if and only if (2^{k-1})M_{k} is
perfect**

k | M_{k} | classification |

1 | 1 | . |

2 | 3 | prime |

3 | 7 | prime |

4 | 15 | composite |

5 | 31 | prime |

6 | 63 | composite |

7 | 127 | prime |

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

**Conjecture:** M_{k} *is prime if k is prime and is composite
if k is composite*.

Here are some factorizations of composite Mersenne numbers:

k | M_{k} | Factorization of M_{k} |

4 | 15 | 3 ^{.} 5 |

6 | 63 | 3^{2} ^{.} 7 |

8 | 255 | 3 ^{.} 5 ^{.} 17 |

9 | 511 | 7 ^{.} 73 |

10 | 1023 | 3 ^{.} 11 ^{.} 31 |

- 3 (=M
_{2}) is a factor of M_{4}, M_{6}, M_{8}and M_{10}. - 7 (=M
_{3}) is a factor of M_{6}and M_{9}. - 31 (=M
_{5}) is a factor of M_{10}.

*if c is a factor of d, then M _{c} is a factor
of M_{d}*

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, M_{11} 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.