Perfect Numbers

Let denote the sum of all positive divisors of n.

 

Proof. \ It is clear that every positive divisor of n has the form

and every such number is a divisor of n. Therefore

The following theorem follows immediately from the above theorem.

 

For example, 6 and 28 are perfect numbers because

Proof. \ It follows from Theorem  that

Now let a be an even perfect number. Suppose that

By Theorem ,

which implies that

Noting that u and are divisors of u, we have that u is a prime and

because is the sum of all positive divisors. We still do not know if any odd perfect number exists, which is a famous difficult problem in number theory. Brent, Cohen and te Riele showed that the lower bound for an odd perfect number is if exists. Brandstein has shown that the largest prime factor is >500000, and Sayers has shown that an odd perfect number has at least 29 prime factors (not necessarily distinct).

1. M. S. Brandstein, New lower bound for a factor of an odd perfect number, No. 82T-10-240, Abstracts Amer. Math. Soc., 3(1982), 257.
2. R. P. Brent, G. L. Cohen and H. J. J. te Riele, Improved techniques for lower bounds for odd perfect numbers, Math. Comput., 57(1991), 857-868.

   M. D. Sayers, An improved lower bound for the total number of prime factors of an odd perfect number, M.App.Sc. Thesis, NSW Inst. Tech., 1986.