Collatz Conjecture

Recently I was reminded of the Collatz Conjecture during a lecture.

It concerns the function

defined by setting

$f_0(n) := 3n+1$ for odd $n$ and $f_0(n) := n/2$ for even $n$.

Collatz Conjecture

For any natural number $n$, the orbit $n$, $f_0(n), f_{0}^{2}(n), f_{0}^{3}(n), ...$ passes through 1

I won’t say that I’ve made much progress on the problem, though I did at several stages think I was near a proof only to realise that I’d then need to probe a subproblem that was incredibly difficult if not impossible.

“Mathematics may not be ready for such problems.” - Paul Erdos

Erdos’ quote here hints at something intrinsic aspect of mathematics–there is always something that we don’t know and something new to be discovered.

Given the years of effort many have spent attempting to prove (or disprove) the conjecture, it certainly seems like it’s unlikely that we will see a proof soon. The conjecture serves to show us just how much we don’t know; even the seemingly simple and intuitive can be beyond current understanding.