Collatz conjecture
(Better known as 3x+1 problem)
I’d say this problem sounds pretty easy.
Consider this sequence: if $a_{i}$ is even, then $a_{i+1} = a_i / 2$; if $a_i$ is odd, then $a_{i+1} = 3 * a_i + 1$. Does all such sequences, starting with positive integer, end up at 1 (contains 1 in them)?
Eventually, no one has ever answered this question (I mean, proven it). This is why it is a conjecture. And, a little suggestion for you – don’t even try to solve it. You’re just wasting your time.
“Mathematics is not yet ripe enough for such questions.” – P. Erdős
Actually, it is enough just to find a counterexample. One counterexample. A number, from which started sequence goes to infinity or gets stuck in the loop (except the only yet found loop, which contains 1). Well, people checked numbers up to 268 – they all ended up to 1.
Also, as you know, there are some problems, that are true and cannot be proven (Gödel’s incompleteness theorems) – maybe this is one of them.
There’s also something to do with Benford’s law (something not really significant), so I will be covering this in the future, I guess.
Eventually, I decided to head into other topics in mathematics and computer science. This problem is not worth putting more time in it. Therefore I will end this up here.
Just last thing I’d like to notice – what if we could make up some kind of “inverse” arithmetic system, in which we could convert/transfer our unsolved math problems, change them in that system, using that systems’ rules (which suits better for the problem), and then convert back to our well-known arithmetic system (something like Taylor series but with it’s own arithmetic rules). I don’t know, maybe someone’s already found something like that, I just have never stumbled upon it. Maybe I should do a research on this idea, before I have invented a bicycle… (As it almost happened with hyperoperations.)
Some links, regarding Collatz conjecture:
https://en.wikipedia.org/wiki/Collatz_conjecture
Audrius
2021-08-31