In mathematics, a perfect number is defined as an integer which is the sum of its proper positive divisors, excluding itself.

Six (6) is the first perfect number, because 1, 2 and 3 are its proper positive divisors and 1 + 2 + 3 = 6. The next perfect number is 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.

Euclid discovered that the first four perfect numbers are generated by the formula 2ⁿ⁻¹(2ⁿ − 1):

for n = 2:   2¹(2² − 1) = 6

for n = 3:   2²(2³ − 1) = 28

for n = 5:   2⁴(2⁵ − 1) = 496

for n = 7:   2⁶(2⁷ − 1) = 8128

Noticing that 2ⁿ − 1 is a prime number in each instance, Euclid proved that the formula 2ⁿ⁻¹(2ⁿ − 1) gives a perfect even number whenever 2ⁿ − 1 is prime.

Ancient mathematicians made many assumptions about perfect numbers based on the four they knew. Most of the assumptions were wrong. One of these assumptions was that since 2, 3, 5, and 7 are precisely the first four primes, the fifth perfect number would be obtained when n = 11, the fifth prime. However, 2¹¹ − 1 = 2047 = 23 · 89 is not prime and therefore n = 11 does not yield a perfect number. 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 alternately end in 6 or 8.

The fifth perfect number (33550336 = 2¹²(2¹³ - 1)) has 8 digits, thus debunking the first assumption. For the second assumption, the fifth perfect number indeed ends with a 6. However, the sixth (8 589 869 056) also ends in a 6. It has been shown that the last digit of any even perfect number must be 6 or 8.

In order for 2ⁿ - 1 to be prime, it is necessary that n should be prime. Prime numbers of the form 2ⁿ − 1 are known as Mersenne primes, after the seventeenth-century monk Marin Mersenne, who studied number theory and perfect numbers.

Two millennia after Euclid, Euler proved that the formula 2ⁿ⁻¹(2ⁿ − 1) 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. This result is often referred to as the "Euclid-Euler Theorem."

Only finitely many Mersenne primes are presently known, and it is unknown whether there are infinitely many of them. Thus it also remains uncertain whether there are infinitely many even perfect 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; also it must have at least 75 prime factors in total, including repetitions), 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.

See also

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.

• Semiperfect number
• Quasiperfect number
• Almost perfect number
• Multiply perfect number
• Hyperperfect number
• Unitary perfect number

External links

• David Moews: Perfect, amicable and social numbers
• Perfect numbers - History and Theory
• Perfect Number - from MathWorld