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![[parent]](http://images.planetmath.org:8080/images/uparrow.png) Viewing Message
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| ``Re: Is following theorem known?''
by silverfish on 2007-06-22 09:52:27 |
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I think you can prove it by making the following handwaving argument work: let n = 2^s -1, for some positive integer s
Then 3n + 1 = 3.2^s - 2, which is divisible by 2 exactly once. So R(n) = 3.2^{s-1} -1
Similarly R^2(n) = 3^2. 2^{s-2} -1....and so on
the point being that the value of R(n) is increasing by a factor of about 3/2 each time. So R^k (n) /n is about (3/2)^k, so long as k < s
If you make s big enough, you can therefore make R^k (n ) /n as big as you like.
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