The 3n+1 problem

Consider the following function:

f(n)={n/2 if n even, 3n+1 if n odd}, n in N.

We ask what happens when we apply f continuously given a certain natural n.

For example, for n=3 we get the sequence: 3, 10, 5, 16, 8, 4, 2, 1. Will any number eventually stabilize on 1? If you try to program a trivial little program that shows you this sequence, you will be convinced that eventually all tried numbers go to 1.

Given that, define a sequence pn=period of n. So in the example above, p3=8. Now define another sequence as follows:

m,n={1≤k≤m: pk=n}. So essentially, m,n counts how many numbers have a given period n in the range 1≤k≤m.

Does m,n converge? More specifically, does limm->∞m,n  exist for given n? What about limn->∞pn?

It turns out the 3n+1 question is an unsolved problem. Given that, one would expect that the second question would be unsolved as well. It appears that m,n diverges. The following are graph plots of m,n versus pn. Successively they are 200,n 400,n 800,n 8000,n and 20000,n. pn ranges on the graphs from 1 to 124, 143, 170, 261 and 278:

200

400

800

8000

20000

Many of these questions are investigated on mensanator's pages.

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