TITLE: Are there specific numbers for which the Collatz Conjecture is proven?
QUESTION [6 upvotes]: Obviously it is proven that all powers of 2 fall to one after applying the rules an arbitrarily large number of times.
But are there some other subsets of natural numbers, for which the Collatz Conjecture has been proven?
As some comments have pointed out there are some proofs by brute force for at least the natural numbers to 2^64. Are there other known subsets, not proved by brute forcing but because of a rigorous mathematical proof?

REPLY [2 votes]: It can be easily shown that all numbers of the form $4k + 1$ will satisfy the conjecture provided that $k$ satisfies it.
$2^kn + 1$ also satisfies the conjecture if $n$ satisfies it, trivially.
Numbers of the form $$\frac{4^m - 1}{3}$$ lead to powers of $2$ and therefore to $1$.
$$\frac{4^mn - 1}{3}$$ leads to $1$ if $n$ leads to $1$.