Re: Project Euclid Problem 26 SOLVED!!!

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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