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6 and 28 are called Perfect Numbers. The proper divisors (the divisors of a number, not including the number itself) of 6 are 1, 2, and 3, and 6=1+2+3. Similarly, the proper divisors of 28 are 1, 2, 4, 7, and 14 and 28=1+2+4+7+14. Are there any other perfect numbers, numbers equal to the sum of their proper divisors? Euclid (in Book IX, Proposition 36) actually showed that if p is prime, and if (2^p)-1 is also prime, then (2^(p-1))((2^p)-1) is perfect (please forgive all the parentheses, but my notation is limited by HTML. 2^p means 2 to the p power). Euler showed that all even perfect numbers are of this form.
Primes of the form (2^p)-1 are called Mersenne Primes. A number of the form (2^n)-1 cannot be a prime unless n is a prime. The nth Mersenne number is Mn=M(n)=(2^n)-1. Many people call M(n) a Mersenne number only if n is prime. The first few Mersenne primes are 3, 7, 31, 127, 8191, etc., which are primes. So the first few perfect numbers are 6, 28, 496, 8128, 33550336, etc. It was quickly shown that M(11), M(23), M(83), M(131) and others were not prime, even though n is prime in those cases. Mersenne apparently conjectured that M(n) was prime for n=2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257, and that M(n) was not prime for any other n below 257. He was shown to be wrong for n=61, 67, 89, 107, and 257. It is relatively easy to test to see if a Mersenne number is prime, and so the largest known prime is often a Mersenne prime.
Here are the 37 known Mersenne primes and perfect numbers (from MathWorld):
| p | M(p) | Perfect(p) | year | source |
| 2 | 3 | 6 | ||
| 3 | 7 | 28 | ||
| 5 | 31 | 496 | ||
| 7 | 127 | 8128 | ||
| 13 | 8191 | 33550336 | 1461 | later mentioned by Reguis and Cataldi |
| 17 | 131071 | 8589869056 | 1588 | Cataldi |
| 19 | 524287 | 137438691328 | 1588 | Cataldi |
| 31 | 10 digits | 2305843008139952128 | 1750 | Euler |
| 61 | 19 digits | 1883 | Pervouchine | |
| 89 | 27 digits | 1911 | Powers | |
| 107 | 33 digits | 1913 | Powers | |
| 127 | 39 digits | 1876 | Lucas | |
| 521 | 157 digits | 1952 | Lehmer | |
| 607 | 183 digits | 1952 | Lehmer | |
| 1279 | 386 digits | 1952 | Lehmer | |
| 2203 | 664 digits | 1952 | Lehmer | |
| 2281 | 687 digits | 1952 | Lehmer | |
| 3217 | 969 digits | 1957 | Riesel | |
| 4253 | 1281 digits | 1961 | Hurwitz | |
| 4423 | 1332 digits | 1961 | Hurwitz | |
| 9689 | 2917 digits | 1963 | Gillies | |
| 9941 | 2993 digits | 1963 | Gillies | |
| 11213 | 3376 digits | 1963 | Gillies | |
| 19937 | 6002 digits | 1971 | Tuckerman | |
| 21701 | 6533 digits | 1978 | Noll and Nickel | |
| 23209 | 6987 digits | 1979 | Noll | |
| 44497 | 13395 digits | 1979 | Nelson and Slowinski | |
| 86243 | 25962 digits | 1982 | Slowinski | |
| 110503 | 33265 digits | 1988 | Colquitt and Welsh | |
| 132049 | 39751 digits | 1983 | Slowinski | |
| 216091 | 65050 digits | 1985 | Slowinski | |
| 756839 | 227832 digits | 1992 | Gage and Slowinski | |
| 859433 | 258716 digits | 1994 | Gage and Slowinski | |
| 1257787 | 378632 digits | 1996 | Slowinski and Gage | |
| 1398269 | 420921 digits | 1996 | Armengaud, Woltman, et al. | |
| 2976221 | 895832 digits | 1997 | Spence | |
| 3021377 | 909526 digits | 1998 | Clarkson, Woltman, et al. |
There may be unknown Mersenne primes less than the last two in the table, as that region has not been searched exhaustively.
It has been conjectured that there are infinitely many Mersenne primes, and therefore infinitely many perfect numbers. No odd perfect number has been found, up to 10^300. The educated guess is that there are no odd perfect numbers.
Addendum:
The book Mystifying Math Puzzles, by Steve Ryan, has this puzzle, labeled "The Perfect Pyramid":
?
28
496
8128
130,816
2,096,128
"What is the missing number that belongs on top of this pyramid, and begins this progressively growing logical number secquence?" Some of the numbers should look familiar. The answer in the back of the book says this:
The missing number is six. All of the numbers on this pyramid are "perfect." A perfect number is a number whose total divisions [proper divisors] equal itself. Ex: 1,2,3 are all divisors of 6 and total six. 1,2,4,7,14 are all divisors of 28 and total 28. There are only six perfect numbers between 1 and 3,000,000.
Of course two of the above numbers are not perfect numbers. And there are only four perfect numbers between 1 and 3,000,000. 130,816 is 2^8(2^9-1), and fails to be a perfect number on two counts: 9 is not a prime, and so 2^9-1 is also not a prime. 11 is a prime, but 2,096,128 (2^10(2^11-1)) fails to be a perfect number because 2^11-1 is not a prime.