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From: johnson542 on 14 Jun 2010 13:17
For a positive integer n, define T_n: Z^+ --> R by

T_n(x)= (2^n * x - 1)/3.

It is easy to see that Collatz Conjecture is true iff every
odd positive integer is equal to

A:= T_(n_k)...T_(n_2)T_(n_1) (1)

for some n_1, ... , n_k in Z^+.

A is an integer only for appropiate (n_1,...,n_k).

Has anyone characterized such k-tuples?
From: TCL on 20 Jun 2010 13:06
On Jun 14, 5:17 pm, johnson542 <johnson...(a)verizon.net> wrote:
> For a positive integer n, define T_n: Z^+ --> R by
>
> T_n(x)= (2^n * x - 1)/3.
>
> It is easy to see that Collatz Conjecture is true iff every
> odd positive integer is equal to
>
> A:= T_(n_k)...T_(n_2)T_(n_1) (1)
>
> for some n_1, ... , n_k in Z^+.
>
> A is an integer only for appropiate (n_1,...,n_k).
>
> Has anyone characterized such k-tuples?

I have been able to do it for k=1,2,3,4,5. (1 and 2 cases are easy). I
am trying to do it now for general k.
Don't know if this (the general k case) has been done by someone
before.
-TCL
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