Any
even perfect number a is of the form a = 2
n-1(2
n-1), where 2
n-1 is a
prime number. While it is difficult to prove that all perfect numbers have this form, it is not very difficult to
prove that these numbers are indeed perfect:
The
divisors of a are 1, 2, 2
2, ..., 2
n-1, and (2
n-1), 2*(2
n-1), 2
2(2
n-1), ..., 2
n-2(2
n-1). The sum of the divisors not containg the (2
n-1)
factor is 2
n-1, and the sum of the other divisors is (2
n-1-1)(2
n-1). Grand total: 2
n-1(2
n-1) = a. Thus a is perfect.