I myself found that working on this seemingly impossible mission, one can anyway learn a lot about numbers, graphs, bases, computers and research strategies. One can also think about how these learnings could be reinvested toward students learning.
I assume you mean the "3x+1" conjecture (aka Collatz conjecture, Hailstones problem).
In 2011, shortly before my retirement, I had a student writing a master's thesis on the topic. After finishing his studies, he couldn't find a job in the area and became an actor, alas. He was quite good and showed original thinking. He made use a.o. of p-adic numbers to attack a slightly more general problem with the function Tl(x) (l > 1 a fixed odd integer)
Tl(x) := l.x+1 if x is odd
Tl(x) := x/2 if x is even.
One of his major results was that for l >= 5 infinitely many integers x will not reach a smaller integer.
I only have a hard-copy of his thesis. I am not an expert myself in these matters. To my knowledge, the original conjecture has been confirmed up to 260. A web site devoted to the subject is kept by Eric Roosendaal (see link).
http://www.ericr.nl/wondrous
Thank you Marcel, Roosendaal's page is an interesting reference. I am working part time on a reversed accelerated version (starting from 1 and considering the subsequences containing only the odd numbers). I managed to create a model of a tree containing all predecessors. I am now studying relations inside that tree and my goal is to show that all odd numbers are in it. My idea is to suppose there is an odd number which does not belong to the tree and get a contradiction by the fact that we would have more than aleph0 odd numbers (in fact, 2^N, which would be aleph1).
You are a courageous man. Way back in the times of the Cold War, the story went that the Syracuse problem was an invention of the Soviet mathematicians to keep Western mathematicians from doing their proper work. Upon hearing the (seemingly) simple question, legions of mathematicians in the West tried their hand at it. Be warned: the problem is addictive.
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