What is currently the highest lower bound for the length of a nontrivial cycle in the Collatz Conjecture?

Question

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.

Answer 1

According to this and, since according to the same site convergence has been computer tested up to $1.003 \times 10^{20}$ by a yoyo@home project (see also this and click "Start" to order from highest to lower number checked), the minimum cycle length should be by now 9,283,867,937. This length is computed counting all sequence steps for both odd and even values, and with one step only from odd $x$ to the following $(3x+1)/2$.

The yoyo guys claim it is 17 billion (search here for "Collatz: Search finished"), however I think Eric Roosendaal is more trustworthy. Maybe this value refers to a double step for odd $x$: $x \to 3x+1 \to (3x+1)/2$.

Answer 2

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)