15
$\begingroup$

The Collatz function $T$ is defined on the set $\mathbb{Z}^+$ of positive integers as: $T(n)=n/2$ if $n$ is even, and $T(n)=3n+1$ if $n$ is odd. Let $T^k$ be the $k$th iteration of $T$. We say $n$ terminates if $T^k(n)=1$ for some $k$.

Let $n$ be an integer of the form $3^{2^k(j-1)}+3^{2^k(j-2)}+\cdots +3^{2^k}+1$

where $k,j\in \mathbb{Z}^+$ and $j$ is odd.

Question: Will $n$ terminate for all such $k,j$?

  • 0
    In my deleted answer I attempted to begin with a simplification of the list of the numbers to be tested, but without making progress in really solving the problem. That simplification can correctly be stated as $ \displaystyle n_k={9^k-1 \over 9^{2^{ \lt k,2 \gt } } - 1} $ where the notation ** means the exponent to which the primefactor *p* occurs in *m* . It seems to me, that this description of the set of numbers involved is somehow better designed for induction or other types of proving (we have now only one parameter *k*, for instance)...2011-12-09

1 Answers 1

1

Not really an answer but there are a few things I'd like to point out:

1) $n_{j,k} = \displaystyle \sum_{i=0}^{j-1} 3^{i \cdot 2^{k}} = \frac{3^{j \cdot 2^{k}}-1}{3^{2^{k}}-1}$

2) Since $j$ is odd, $n_{j,k}$ is also odd.

3) Thus, $T(n_{j,k})=3 \cdot n_{j,k}+1$ and $T^{2}(n_{j,k})=\frac{1}{2}(3 \cdot n_{j,k}+1)$, since $3n+1$ is even for all $n$.

4) $n_{j,k}$ has an interesting form when viewed in base 3. For example,

IntegerDigits[n[7,3],3] {1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1} 

That is, for $n_{j,k}$ we get a 1 followed by $j$ zeros, etc.

5) Here is a plot of the lengths of the sequences $T^{k}(n_{j,k})$ for $1 \le j \le 16$ and $1 \le k \le 8$, although for these sequences I used a highly reduced version of the Collatz function where if $n$ is even, $T(n)=n/2^{\kappa}$, where $\kappa=\max\{k : \, 2^{k}|n \}$:

enter image description here

Even though these numbers have a certain form that would seem to make it easier to prove they have a downward trajectory to 1, after looking around a little bit I really didn't seem to see any patterns. $\{n_{j,k} : j \, \text{odd}, k \in \mathbb{N}^{+}\}$ is a fairly "large" set of integers, so this question might really be similar in difficultly to the full conjecture.