## Perfect Numbers

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.

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.