Here's an old post by terry tao explaining why the collatz conjecture is hard, and giving a heuristic argument that it's unlikely to be solved with purely elementary techniques. The essay is surely worth reading if you want to make an attempt on that problem!
https://terrytao.wordpress.com/2011/08/25/the-collatz-conjec...
Re:The Erdos Paradox
by cjohnson318 on Thursday January 03, 2019 @11:03PM
(#18821880)
I don't think that a list of historically unsolved problems in math, and the Collatz Conjecture in particular, count as low-hanging fruit. The fact that they have remained unsolved for so long suggests that they are rather high-hanging.
Heh, fair enough. Was thinking more things that might not need advanced math to solve. Or at least to understand the problem.
Re:The Erdos Paradox
by intuitionist on Thursday January 03, 2019 @10:45PM
(#18821811)
For another, more recent counterexample see the Hirsch conjecture: https://en.m.wikipedia.org/wiki/Hirsch_conjecture
No way you'd get research money for this, but an amateur can sit and poke away at it.
These are all unsolved: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_m..., and plenty look like they may be solvable without advanced math, just someone dedicated and a unique approach.
Particularly the Collatz Conjecture, afaik basically zero progress has been made on it, no "advanced" math topics are known to apply at all. So perhaps a unique angle is all you need. (Or maybe it's false, in which case all you need is a counterexample. Sometimes it happens, see e.g. https://www.ams.org/journals/mcom/1988-51-184/S0025-5718-198...)