What level of mathematics do I need to study the Collatz Conjecture?

Question

I recently came across the Collatz Conjecture and I'm really intrigued by its tautological simplicity and complexity.

I'm under no illusions that I can make any progress with a proof for it but I really want to study it more and be able to understand attempts at proofs.

I've studied maths to A2 level (albeit a few years ago now), so any advice as to how much more I'd need to study, in particular what areas? Any tips as to where I could learn more and the best way to go about it would be really appreciated.

Thanks!

Answer 1

The beauty of this problem is that it can be studied with any level of mathematics. Solving it however, is a different matter.

The fact that so many have tried and failed proves the following:

If proof with elementary mathematics is possible, then it requires some exceptional feat of logic, reasoning, or intuition, which has eluded so many people before.

If proof with more difficult mathematics already in existence is possible, then it still probably requires some exceptional feat of logic, reasoning, or intuition, which has eluded so many people before.

However... I think it was Erdos who said, he didn't believe the mathematics to solve this problem yet exists.

There are many methods which can be used to tackle this problem. A proof is more likely to come out of an advanced field of number theory, such as the p-adics and modular elliptic curves.

If however there is some field of relatively elementary mathematics you wanted to study, which is more likely to yield a result, it is the field of modular arithmetic, because the problem derives its complexity out of the moire that arises out of mod 2 and mod 3 arithmetic.