Perfect number
A positive integer is called a perfect number if it is the sum of its proper divisors.
The first four perfect numbers are:
These were the only perfect numbers known to ancient mathematicians. Some other perfect numbers include:
Even Perfect Numbers
is an even perfect number if and only if , where is a prime number of the form for some . Primes of the form are called Mersenne primes.
Proof
It is conjectured that there are are infinitely many Mersenne primes and hence infinitely many even perfect numbers. No odd perfect numbers are known, and any that do exist must be greater than . It is conjectured that there are none. No one has been able to prove or disprove these conjectures.
Numbers which are not perfect may be either deficient numbers or abundant numbers.