http://en.wikipedia.org/wiki/Sociable_number

I was reading the wiki on Sociable numbers and I found it interesting that there are no known social numbers of order 3 (i.e. there are three numbers in the cycle). Do you think there's a way to prove that social numbers do or do not exist for any known order?

Also, the wiki talked about the theorizing the possibility that all numbers (positive integers) are either part of a social cycle or terminate at 1 by summing up proper divisors to get numbers in the next sequence. While this seemed kind of mind blowing — that perhaps there might be a counterexample to this of a sequence of composite numbers that never reach a prime, since all primes determine the next number in the sequence to be 1 — it also reminded me of a similar conjecture that i enjoy: the Collatz Conjecture.

The Collatz Conjecture makes a statement about a certain sequence. Take any positive integer to be the first term of a sequence and consider the following operations to determine the next number in the sequence. If the number is even, divide it by two, but if it is odd, multiply it by three and add one. The Collatz Conjecture says that for any sequence starting at any positive integer, this sequence will always reach 1.

Pretty crazy, huh? I think so, anyway. Paul Erdős said of the problem, "Mathematics is not yet ready for such problems." He offered $500 for its solution, which if you know Paul (or know of him), is quite the sum.