Perfect number

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

Noticing that is a prime number in all computations, Euclid proved that the formula gives a perfect even number whenever is prime.

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.

The fifth perfect number has 8 digits, thus debunking the first assumption. For the second assumption, the fifth perfect number indeed ends with a 6. However, the sixth also ends in a 6. It hasn't been proven that all perfect even numbers always end in a 6 or 8, but so far, the first thirty do. Also, each of these perfect numbers that ends in 8, really ends in 28.

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 10³⁰⁰. 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 10⁷, two prime factors greater than 10⁴, 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.

Also see

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

External links

• Encyclopedia of Integer Sequences: A000396
• David Moews: Perfect, amicable and social numbers