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From: Physfitfreak <Physfitfreak@gmail.com>
Newsgroups: comp.os.linux.advocacy
Subject: Re: Project Euclid Problem 26 SOLVED!!!
Date: Mon, 11 Mar 2024 16:17:08 -0500
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On 3/11/2024 3:40 AM, Farley Flud wrote:
> On Sun, 10 Mar 2024 13:46:06 -0500, Physfitfreak wrote:
> 
>>
>> The second image is also 0 to 1000, not 10000.
>>
> 
> The second image is now corrected:
> 
> https://i.postimg.cc/FmKpsHzY/prob26-10k.png
> 
> Again, the resolution is 10000x1200 and it should be downloaded
> and viewed off-line.
> 
> There is a clear upward, seemingly linear trend and this will
> undoubtedly continue to infinity.
> 
> The educational system only describes repeating decimals
> briefly because an in-depth analysis is beyond its resources.
> 
> But now we have CAS and we can examine repeating decimals
> with ease.
> 
> Maxima can calculate this series up to a billion or more,
> but it would take some time.  I need to check out Pari,
> which is a C program, and thus much faster:
> 
> https://pari.math.u-bordeaux.fr/
> 
> The relevant Pari function would be znorder(x, {o}):
> 
> https://pari.math.u-bordeaux.fr/dochtml/html/Arithmetic_functions.html
> 


Now 4 distinct trends can be discerned, with 4 different slopes. I can 
only guess right now, but it may have something to do with the fact that 
from 1 to 9 as denominator, there are four cases of repeated decimals 
(3's, 6's, 142857's and 1's).

What are larger integer intervals anyway. They're like the 1 to 9 
interval, only magnified to show higher resolutions, allowing more 
numbers in between 1 and 9 to be treated as integers. This might be the 
reason behind those 4 trends.

And if that's the case, then the same plot, but for 1 to 100000 or 
higher interval, should still show only 4 trends, and not more.



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