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When I look at the encyclopedia of integers, it is clear that this grows even though it doesn't grow continuously.

It seems to me that it should be possible to establish a lower bound based on identifying different cases where $m$ is the minimum power of $2$ greater than $3^n$:

Case 1: $2^m > 3^n$ and $2^{m+1} > 3^{n+1} > 2^m$

In this case, $2^m > \dfrac{3}{2}3^n$ so $2^m - 3^n > \dfrac{3^n}{2}$

Case 2: $2^m > 3^n$, $2^{m+2} > 3^{n+1} > 2^{m+1}$ and $2^{m+3} > 3^{n+2} > 2^{m+2}$

In this case, $2^m > \dfrac{9}{8}3^n$ so $2^m - 3^n > \dfrac{3^n}{8}$

I am able to define the remaining case but I am not able to find a pattern.

Case 3: $2^m > 3^n$, $2^{m+2} > 3^{n+1} > 2^{m+1}$, and $2^{m+4} > 3^{n+2} > 2^{m+3}$

In this case, I am not able to find a generalizeable lower bound. I'll give an example. Case $3$ holds for $m=8, n=5$. In this case, eventually, I find that $2^{m} > \dfrac{3^{12}}{2^{19}}3^n = \dfrac{531,441}{524,288}3^n$

When I check the next example of case 3 at $m=16, n=10$, I get $2^{m} > \dfrac{3^7}{2^{11}}3^n = \dfrac{2187}{2048}3^n$

Is there a straight forward way to find a lower bound for Case 3? Is this an open problem?

Larry Freeman
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    Hmm, instead of writing $2^m-3^n \gt \frac{3^n}a$ you might like to write $ {2^m-3^n\over 3^n } \gt \frac1a$ and you can calculate this easier for large $n$ when convert to ${ 2^m\over 3^n } -1 \gt \frac1a$ and then to $ 2^{1- { n \gamma }} -1 \gt \frac1a$ where $\gamma = \log_2(3)$ . Then your typology/pattern might become more visible if you consider that $n$ from the convergents of continued fraction of $\gamma$. For instance, if you compute $ w_n= 2^{1- { n \gamma }} -1 $, then $n \cdot w_n$ seems to give nicely bounded values with possibly better discernable patterns. – Gottfried Helms Jun 07 '21 at 21:43
  • Thanks, @gottfriedhelms I was quite surprised by the content of Collag3n's answer. Lots to think about! I appreciate your observations! Would you know if there is an English article covering the Rhin bound? – Larry Freeman Jun 07 '21 at 22:47
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    I do not know more than the articles of J. Simons & B. deWeger on 2-cycles and m-cycles where they use the Rhin-bound. Unfortunately I don't speak french so I've no access to/understanding of the Rhin's original article. Simons resp deWeger have homepages where they provide they articles, I think they should also be available on arXiv. – Gottfried Helms Jun 08 '21 at 03:25
  • Thanks @gottfriedhelms. The Simons & DeWeger is a great paper. I'll start there. – Larry Freeman Jun 08 '21 at 03:35
  • Ah, well, perhaps you like also this older discussion of mine about the subject: http://go.helms-net.de/math/collatz/2hochS_3hochN_V2.htm I look at the tendency of that differences up to very large $n < 10^{100000}$ and have made pictures of it. Unfortunately the text has become somehow uneasy due to updates without coolness-in-mind ;-) Perhaps it is a good moment to look for a complete rewrite to make it more straightforward... But for curiosity it should serve well, I think... – Gottfried Helms Jun 08 '21 at 07:36

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With $m=\lceil n \log_23\rceil$, bounds for $2^{\lceil n \log_23\rceil}-3^n$ can be found here: Show that $2^n<2^{\lceil n \log_23\rceil}-3^n<3^n-2^n$

Collag3n
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