10 Perfect numbers
In mathematics the pursuit of perfection has led its aspirants to different places. There
are perfect squares, but here the term is not used in an aesthetic sense. It’s more to
warn you that there are imperfect squares in existence. In another direction, some
numbers have few divisors and some have many. But, like the story of the three bears,
some numbers are ‘just right’. When the addition of the divisors of a number equals the
number itself it is said to be perfect.
The Greek philosopher Speusippus, who took over the running of the
Academy from his uncle Plato, declared that the Pythagoreans believed that 10
had the right credentials for perfection. Why? Because the number of prime
numbers between 1 and 10 (namely 2, 3, 5, 7) equalled the non-primes (4, 6, 8,
9) and this was the smallest number with this property. Some people have a
strange idea of perfection.
It seems the Pythagoreans actually had a richer concept of a perfect number.
The mathematical properties of perfect numbers were delineated by Euclid in the
Elements and studied in depth by Nicomachus 400 years later, leading to
amicable numbers and even sociable numbers. These categories were defined in
terms of the relationships between them and their divisors. At some point they
came up with the theory of superabundant and deficient numbers and this led
them to their concept of perfection.
Whether a number is superabundant is determined by its divisors and makes a
play on the connection between multiplication and addition. Take the number 30
and consider its divisors, that is all the numbers which divide into it exactly and
which are less than 30. For such a small number as 30 we can see the divisors
are 1, 2, 3, 5, 6, 10 and 15. Totalling up these divisors we get 42. The number
30 is superabundant because the addition of its divisors (42) is bigger than the
number 30 itself.
The first few perfect numbers
A number is deficient if the opposite is true – if the sum of its divisors is less