# Collatz stopping-time and Poisson distribution, and connection to other problems?

I read many threads about Collatz here - so don't worry, this is no attempt to any proof, just asking about a curious fact:

This graph gives the stopping-time of Collatz sequences up to $n=10^8$ (source: http://en.wikipedia.org/wiki/File:CollatzStatistic100million.png ) and it's distribution looks very similar to a Poisson distribution.

Is there some known reason why the Collatz-sequence stopping time behaves like a poissonian distribution?

What are the connections to other mathematical problems: Does the Collatz-conjecture imply other conjectures, or do other conjectures imply the Collatz-conjecture?

Thank you!

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By the way, that doesn't closely resemble a Poisson distribution. Poisson distributions have a variance equal to the mean, so a standard deviation equal to the square root of the mean. The mode looks about like $13^2$. Is about $95\%$ within $26$ of the mode? No, it looks like the variance is several times higher than the mean. – Douglas Zare May 12 '13 at 23:47
Did you mean "threads" or was it really "threats"? In both cases, however, the question remains the same :-) – Suvrit May 13 '13 at 1:16

## 1 Answer

As to the observed distribution of total stopping times for integers $n \leq 10^8$, I think heuristically this can be explained quite well by the obvious stochastic model (multiply $n$ by $3/2$ or $1/2$, each with probability $1/2$, repeat this until the number gets $\leq 1$ and count the number of steps this takes). For literature on such stochastic considerations, see Lagarias' annotated bibliography on the conjecture (http://arxiv.org/abs/math/0309224). Proving anything is of course quite a different task!

The Collatz conjecture can be formulated in quite a number of equivalent ways, see also Lagarias' bibliography. Though so far the conjecture is by far not as well-embedded into known parts of mathematics as for example the Riemann Hypothesis. Namely I am not aware of, say, important 'theorems' which are proved up to the Collatz conjecture. Nevertheless, I think the fact that an easy-to-state question like the Collatz conjecture seems so intractable suggests that certain important things are not understood so far. Hence I don't think the conjecture will seem as isolated as now for all future -- but of course this is my personal view.

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Thank you for the answere. Indeed your explanation with the simple statistical model gives the same result for big n (and interesting deviations for smaller n - just tested it with mathematica). Very interesting that it follows the rules of random events for big numbers - did not expect. – MarkusWave May 12 '13 at 21:44
So if you use all the $2$-adic integers, with its Haar measure, then it would have a nice distribution. But corresponding properties upon restricting to the rational integers (a set of probability zero) are not simple to prove, or even (so far) possible to prove. – Gerald Edgar Apr 12 at 0:13