A **perfect number** is an integer which is the sum of its proper positive divisors (factors), not including the number itself.

Thus, 6 is a perfect number, since 1, 2 and 3 are its proper positive divisors, and 6 = 1 + 2 + 3. The next perfect number is 28, as 28 = 1 + 2 + 4 + 7 + 14. The next perfect numbers are 496 and 8128. These first four perfect numbers were the only ones known to the Ancient Greeks.

The Greek mathematician Euclid discovered that the first four perfect numbers are generated by the formula :

- For
*n*= 2, = 6 - For
*n*= 3, = 28 - For
*n*= 5, = 496 - For
*n*= 7, = 8128

Ancient mathematicians made many assumptions about perfect numbers based on the four numbers they knew. Most of the assumptions were wrong. One of these assumptions was that since *n* is also prime for the four computations and the first four primes to boot, they thought that the fifth perfect number would be obtained when *n* = 11, the fifth prime. It didn't. For when *n* = 11, is not prime. Two other wrong assumptions were:

- The fifth perfect number would have five digits since the first four had 1, 2, 3, and 4 digits respectively.
- The perfect numbers would always end in 6 or 8 and that they would alternate repeatedly.

It's true that *n* should be a prime but it does not mean that is automatically a prime. Nowadays, prime numbers generated by the formula are known as Mersenne primes, after the seventeenth-century monk, Marin Mersenne, who studied number theory and perfect numbers.

Two millennia after Euclid, Leonhard Euler proved that the formula will yield all the even perfect numbers. Thus, every Mersenne prime will yield a distinct even perfect number—there is a concrete one-to-one association between even perfect numbers and Mersenne primes.

Only finitely many Mersenne primes (hence even perfect numbers) are presently known. It is unknown whether there are infinitely many of them. See the entry on Mersenne prime for additional information concerning the search for these numbers.

It is unknown whether there are any odd perfect numbers. Various results have been obtained, but none that have helped to locate one or otherwise resolve the question of their existence. It is known that if an odd perfect number does exist, it must be greater than 10^{300}. Also, it must have at least 8 distinct prime factors (and at least 11 if it is not divisible by 3), and it must have at least one prime factor greater than 10^{7}, two prime factors greater than 10^{4}, and three prime factors greater than 100.

Considering the sum of proper divisors gives various other kinds of numbers. Numbers where the sum is less than the number itself are called *deficient*, and where it is greater, *abundant*; these terms, together with *perfect* itself, come from Greek numerology. A pair of numbers which are the sum of each other's proper divisors are called *amicable*, and larger cycles of numbers are called *sociable*. A number which is amicable to itself is perfect.

- Abundant number
- Deficient number
- Quasiperfect number
- Almost perfect number
- Multiply perfect number
- Hyperperfect number