TITLE: What is currently the highest lower bound for the length of a nontrivial cycle in the Collatz Conjecture?
QUESTION [5 upvotes]: We know that there are two possibilities to disprove the Collatz Conjecture.

We find a nontrivial cycle.
We find a sequence that diverges to $\infty$

A non-constructive disproof is imaginable as well. I am particular interested in the cycles that have been ruled out.
I read the questions and answers about the Collatz conjecture in MSE. I would like to learn. 

What is the longest cycle that has been ruled out until now? For example, is it possible to prove that there is no cycle of $10 ^ {1000} $  (or otherwise)?

I present an example for negative integer number that best describes the definition of the length of the cycle. 

$$17 → −50 → −25 → −74 → −37 → −110 → −55 → −164 → −82 → −41 → −122 → −61 → −182 → −91 → −272 → −136 → −68 → −34 → −17$$

So, we have $\large 7$ odd-value cycle length.
But here, Collatz Conjecture doesn't include negative number.

REPLY [5 votes]: The following link asserts an improvement on Eliahou’s lower bound on cycle lengths: p. 13 in the slides of T. Ian Martiny’s talk
The author gives a lower bound  for a cycle length to be 10, 439, 860, 591. 
Update (redux): I read the slides too hastily. The result rules out ranges of cycle lengths: a cycle length must admit the representation 
$$
630 138 877a + 10 439 860 591b + 103 768 467 013c
$$
where $a,b,c$ are non-negative integers, $b>0$, and $ac=0$.
(Thanks--again--for the clarification and keeping me honest, G. Helms)