Perfect Numbers

Perfect numbers were given their name by the ancient Greek mathematiticians, who mixed number theory with mysticism. A perfect number is a number that is equal to the sum of all of its (positive) divisors, excluding itself.

6 is the first perfect number because 6 = 1 + 2 + 3.
28 is also perfect, because 28 = 1 + 2 + 4 + 7 + 14.
The third perfect number is 496, and the fourth is 8128. All of these were found by the ancient Greeks. The fifth perfect number, 33,550,336, is much larger than the first four. It was first mentioned in a fifteenth-century manuscript.

The Greek mathematician Euclid discovered a pattern that allowed him to show that 496 and 8128 are perfect numbers. He proved that, if 2 ^k - 1 (where k is a positive integer) is prime, then 2 ^k-1 × (2 ^k-1) is perfect. Primes of the form 2 ^k - 1 are called Mersenne primes. For example, 8128 = 2 ⁶ × (2 ⁷-1). All of the perfect numbers generated by this method are even. In the 18th century, Leonhard Euler proved that all even perfect numbers are of Euclid's form.

On the other hand, odd perfect numbers are a mystery to mathematicians. No-one has found an odd perfect number, but no-one has been able to prove that all perfect numbers are even. There are many conditions that an odd perfect number must satisfy. () Euler proved that an odd perfect number must be of the form p ^a x q ^b x r ^c x ..., where p, q, r, etc. are of the form 4n + 1, a is of the form 4n + 1, and b, c, etc. are all even. More recently, it was shown that an odd perfect number must have at least eight distinct prime factors. If 3 is not one of those factors, at least eleven distinct factors are required. It must also be divisible by a prime power greater than 10 ¹⁸. The greatest prime factor must be greater than 300,000 and the second largest must be greater than 1000. Any odd perfect less than 10 ⁹¹¹⁸ is divisible by the sixth power of some prime. In Richard Guy's Unsolved Problems in Number Theory, he states that the lower bound for an odd perfect has been raised (as of 1981) to above 10 ²⁰⁰, although he writes that there is some scepticism about the later proofs. I believe that this scepticism may be due to the fact that these proofs were computer-assisted. Remember that Guy was writing only a few years after the four-colour proof controversy. It is possible, though, that some number may just satisfy all of the conditions.

Currently, about 39 perfect numbers are known, although the Great Internet Mersenne Prime Search discovers new ones from time to time.

Amicable numbers are similar in principle to perfect numbers.