A number is said to be
perfect if it is equal to the sum of all its divisors, excluding itself. Perfect numbers have been studied by ancient Greek mathematicians including
Euclid. They are closely related to
Mersenne primes.
Euler has
proved that all even perfect numbers can be written as:
for all prime
p associated with a Mersenne prime. The last expression above hints that even perfect numbers must be
triangular. The first few even perfect numbers, when
p = 2, 3, 5, 7, 13, are:
6, 28, 496, 8128, 33550336.
It is not known whether odd perfect numbers exist.