Re: Project Euclid Problem 26 SOLVED!!!

On 3/12/2024 9:58 AM, Nuxxie wrote:

On Mon, 11 Mar 2024 08:40:32 +0000, Farley Flud wrote:
 

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The second image is now corrected:
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https://i.postimg.cc/FmKpsHzY/prob26-10k.png
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 This is definitely a global first.  Such n plot has never before appeared
anywhere in the history of civilized man.  Never.
 And it's all due to GNU/Linux/FOSS.
 

It is a plot nice to have access to for some enthusiasts. Make sure it doesn't get lost. Place is somewhere on the web where related enthusiasts can permanently access it at will.
It provides a rare peep into how things are for larger integers at the denominator of rational numbers. Not derived, but actually computed.
Then, somebody should derive such features by math alone, not computing. At least the main features. It could be publishable too, if not derived before. But I doubt it's something novel. Math people have done all sorts of investigations, many of them even long ago.
Here are some of the facts it depicts.
- the plot shows, with certainty, that strange surprises aren't in store (this is why I wanted to see it)
- it shows firm limits to the sizes of repeated decimals' length, that depend only on the size of the integer in the denominator
- it shows four trends (so far) as one goes for larger integers in denominator
- it hints on endless repeating structures inside the 1 to 9 interval, as we go deeper and deeper into higher and higher resolution to deal with numbers existing therein
All the four features above should hold for any base, not just 10. There's really nothing special about base 10. The base can be chosen as any positive real number. In fact, it would be nice to have the same plot, but for number system in base e :-)
The reason I like this "e" base is, there must be a reason (which I still don't know or remember) that when used as base in logarithms, makes them called  "natural" logarithm. There must be some special significance to it.
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