*Another one of the irregular breaks from the personal drama... it's a math
post, this time around. Have fun!*
a number is called perfect if it is equal to the sum of its own proper
divisors, in other words, all the numbers that divide into it exactly,
including one, but excluding the number itself. For example, the number 6
has divisors 1, 2, 3. 1+2+3=6, so 6 is the smallest perfect number. The next
one after that is 28. A long time ago, a proof was found that every
*even *perfect
number must equal a Mersenne prime, multiplied by the power of 2 smaller
than it. 28, for instance, is equal to (2^2)*(2^3-1), where 2^3-1 = 7 is a
Mersenne prime. At the time of writing there are 47 known Mersenne primes;
hence 47 known perfect numbers, the largest of which having almost *26
million digits*. If even perfect numbers seem rare, it is still not even
known if an *odd *perfect number exists or not, although the many theorems
constructed about its existence suggest that, should one exist it would be
very large and of a very special form indeed. It seems one might construct a little game when it comes to this magical
"sum of proper divisors" function. Starting with any number, we could take
the sum of its proper divisors; then continue with that number; then
continue with that number, and so on. Who knows, perhaps this way we might
accidentally stumble on a perfect number, or perhaps something else. Perhaps
it is best to illustrate by an example. Let us start with the number 20. Its
proper divisors are 1, 2, 4, 5, 10, with sum 22. We will call a number like
20, whose divisor sum is larger, an *abundant number*. 22's divisors are
just 1, 2, and 11, with divisor sum 14. We will call a number like 22, whose
divisor sum is smaller, a *deficient number*. Perfect numbers exist
somewhere between these two extremes. Let's carry on. 14's divisors are 1,
2, and 7, with sum 10. 10's divisors are 1, 2, and 5, with sum 8. 8's
divisors are 1, 2, and 4, with sum 7. And 7's only divisor is 1. End of the
road - 1 doesn't have any divisors except for itself. It takes a little to prove an *iterative* process like the one above,
applying the same function over and over again, has only a limited number of
outcomes. The sequence may stop, such as when it reaches a number which we
can no longer apply, such as 1; the sequence may eventually repeat, looping
over the same numbers again and again in a cycle; or the sequence may go on
forever, never revisiting any previous entries and inevitably visiting
numbers that grow ever larger. It is intriguing to see if there are other
possibilities other than numbers immediately repeating, such as the perfect
numbers, or numbers ending in 1. Perhaps, even if the perfect numbers are
rare, this little trick has something else to offer. 220 is an interesting number. Its sum of divisors is 284, and... perhaps
you've guessed it, summing the divisors of 284 gets us back to 220. Such a
pair is called an *amicable pair* - almost like best friends of numbers,
perhaps? While not easy to find, they are certainly more common than perfect
numbers, and many impressive examples are known to
exist.
220, 284 is merely the smallest such pair, and was known to our old friend
Pythagoras. Discoverers range from Euler and Fermat of years gone by, to
researchers right up to the present day. One helpful thing about amicable
pairs is there is a handy formula for finding the sum of all divisors of a
number (including the number itself), provided you know the number's prime
factorization. Replace each term *p^a* in the number's factorization with *
(p^(a+1))/(p-1)*, and the product now gives the sum of all divisors. An
amicable pair is two numbers *a* and *b*, both of whom have the same sum of
all divisors, *a+b*. In the example, the all-divisors sum of 220 and 284 are
both equal to 504. This concept can be even extended further, you can find
amicable triplets, whose all-divisor sums are equal to the sum of all three
numbers, and so on. Going back to the idea of iterating the sum of proper divisors function over
and over, it turns out we can find some remarkable numbers which eventually
cycle back to themselves in an *amicable chain*. Beginning with 12496, for
example, takes us through 14288, 15472, 14536, 14264 before returning to
12496 on the fifth step. Starting with 14316, it takes 28 steps before we
get back to the original number. Are there any longer chains than that? And,
even if you do not find a chain, are there really any numbers for which the
sequence would go on forever? It's conjectured there are none.
