Abstract

The following results concerning even perfect numbers and their divisors are proved: (1) A positive integer n of the form 2p−1(2p−1), where 2p−1 is prime, is a perfect number; (2) every even perfect number is a triangular number; (3) τ(n)=2p, where τ(n) is the number of positive divisors of n; (4) the product of the positive divisors of n is np; and (5) the sum of the reciprocals of the positive divisors of n is 2. Values of p for which 30 even perfect numbers have been found so far are also given.