I know the Collatz conjecture is dangerous ground and laymen should not be playing here. Please bear with me.
I'm interpreting Erdös' sentence as, we are lacking tools to tackle such kind of problems. My guess is the problem was tried to be solved with methods of most/all fields and no progress was made.
In 2019 Alex Kontorovich outlined an idea to construct a counterexample, although the meat is still missing. (Not sure whether this is his or an older idea)
My question is:
- Is Erdös's claim still valid? And was my interpretation of his sentence right?
- Nobody knows this so please provide a reasoning: What are methods/fields which are most promising to be close to a solution?
- Are there similar ideas like Alex Kontorovich's one?
The Collatz conjecture is closely linked to the prime factorization of consecutive integers, i.e., the ''missing link'' between additive and multiplicative number theory.
In particular, we still don't know an explicit relation between the prime factors of the positive integer $n+1$ given the prime factorization of $n$.
You can find more info about this here.