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In mathematics, a perfect number is an integer which is the sum of its proper positive divisors, excluding itself.
Thus, 6 is a 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 (sequence A000396 in OEIS). 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 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 (<math>33550336=2^{12}(2^{13}-1)<math>) 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. (That the last digit of any even perfect number must be 6 or 8 is not difficult to show.)
In order for <math>2^n-1<math> to be prime, it is necessary that <math>n<math> 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, Leonhard 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.
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 69 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.
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.
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