perfect number. It is easy to check it really is the sum of its divisors: 496 = 1 +
2 + 4 + 8 + 16 + 31 + 62 + 124 +248. For the next perfect numbers we have
to start going into the numerical stratosphere. The first five were known in the
16th century, but we still don’t know if there is a largest one, or whether they go
marching on without limit. The balance of opinion suggests that they, like the
primes, go on for ever.
The Pythagoreans were keen on geometrical connections. If we have a perfect
number of beads, they can be arranged around a hexagonal necklace. In the case
of 6 this is the simple hexagon with beads placed at its corners, but for higher
perfect numbers we have to add in smaller subnecklaces within the large one.
Mersenne numbers
The key to constructing perfect numbers is a collection of numbers named
after Father Marin Mersenne, a French monk who studied at a Jesuit college with
René Descartes. Both men were interested in finding perfect numbers. Mersenne
numbers are constructed from powers of 2, the doubling numbers 2, 4, 8, 16,
32, 64, 128, 256,.. ., and then subtracting a single 1. A Mersenne number is a
number of the form 2n − 1. While they are always odd, they are not always