I've been obsessing on this problem for a couple weeks. While thinking on it I made some observations on my own, so it's worth a post on my blog. Note that I'm not a mathematician, so there's more English than Math here (which, for people like me, could in fact make it easier to read). I don't have proofs for the most interesting observations I made, and they might be known already for the trained mind.
If you studied the Collatz conjecture, then you might want to jump to the TL;DR.
The Collatz conjecture defines a sequence of positive integers that starts with any number $n > 1$, and at each step, in order to get the next number, you apply this simple logic, depending on the parity of the current number $n$:
- When $n$ is even, the next number is $n / 2$
- When $n$ is odd, the next number is $3n + 1$
The conjecture states that any such sequence eventually reaches $1$ (of course, we could continue after $1$ but it will loop). For an example, let's start with $n = 6$: 6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1.
[ read more... ]