What’s the story on perfect numbers?

Mathochist replies:

Perfect numbers are a holdover from the days of the Pythagoreans, when mathematicians were mystics as much as anything else and put a lot more stock in coincidence.

Start with a number. Find all the numbers that divide it evenly. Add them all up (other than the number itself), and sometimes you’ll get your original number back again. These are the perfect numbers. For example, 6 is evenly divisible by 1, 2, and 3. 1+2+3 = 6. Others are (see reference 1):

28

496

8,128

33,550,336

137,438,691,328

2,305,843,008,139,952,128

2,658,455,991,569,831,744,654,692,615,953,842,176

191,561,942,608,236,107,294,793,378,084,303,638,130,997,321,548,169,216

These are the first 9 of the 38 known perfect numbers. Luckily, we don’t have to factor and add up the divisors to check these. Euler proved in 1747 that even perfect numbers all have the form 2^(n-1) * (2^n-1), where 2n-1 is a kind of prime number called a Mersenne prime. Also, given a Mersenne prime, applying the formula will always give a perfect number. That is, Mersenne primes and even perfect numbers go hand in hand.

So what about odd perfect numbers? Euler also showed that any such number must be of the form (4a+1)4^(b+1)c^2, where b and c are numbers and 4a+1 must be prime. Unfortunately, as of 1991, Richard Brent and Graeme Cohen have established a lower bound of 10300 for odd perfect numbers. That is, if there are any, they’re at least that big. Cohen also showed in 1998 that every odd perfect number (again, if they exist) has a prime factor bigger than 106. Douglas Iannucci showed in 1999 and 2000 that the second and third largest prime factors are at least 10,000 and 100, respectively.

These days, we usually say that the sum of all the divisors (including the number itself) of a perfect number is twice the number (i.e., for the perfect number 6, 1+2+3+6=12). If the sum is greater than the number, we call the number “abundant”. If it’s less, we says it’s “deficient.” Like “perfect,” these terms date to the Pythagoreans.

In the 17th century, mathematicians like Fermat started coming up with variations on the perfect-number theme. A pseudoperfect number is the sum of some or all of its divisors. Quasiperfect numbers have a divisor sum of 2n+1; almost perfect numbers have a divisor sum of 2n-1. Then there are pluperfect numbers, which Descartes studied (or had studied, perhaps I should say)–those whose divisors sum to some other multiple of n than 2. There are also amicable numbers, sociable numbers, harmonic numbers, hyperperfect numbers, infinitary perfect numbers, unitary perfect numbers, super unitary perfect numbers, superperfect numbers, sublime numbers, and weird numbers–all of which satisfy some quirky condition or another. For example, sublime numbers have a perfect number of divisors, and the sum of their divisors is itself perfect. The only two yet known are 12 (1+2+3+4+6+12=28, which is perfect, and there are 6 divisors, which is also perfect) and 698,655,567,023,837,898,670,371,734,243,169,822,657,830,773,351,885,970,528,324,860,512,791,691,264.

So, why are only 38 perfect numbers known after all this time? Well, other than playing around with them, there’s just not a lot that they’re good for. I mean, if having a good list of perfect numbers could help split the atom or cure cancer you can bet mathematicians would be on the problem like Dopers on a grammatical error. As it is, we’ve just got more important things to work on, like whether you can untie this or that knot.

As for the math mafia, I’d tell you but the Don would have my head.

References

1. Sequence A000396 of the On-Line Encyclopedia of Integer Sequences, accessed May 3, 2005

Mathochist

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