畷 (na-wa-te) 60 Stars Astrology Season Holiday extra 7 of 3 : Combining Numbers to Measure Something Part 6 "The End of the Journey of Fermat Numbers"

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60 Stars Astrology  Season Holiday Extra 7of 3

Combinations of Numbers for Measuring Something part 6

"The End of the Journey of Fermat Numbers"

1.  Well, Tanu-chan set out on a journey of Fermat numbers, but even while making the table (index) of contents for learning the progression, I continued my train journey diligently.

 
But, in the end, I still didn't understand what the true purpose of Fermat numbers was that Fermat had seen.


.....Of course, I regret that I was a bit too ambitious for a small animal. 
But I had no evil intentions.




2. I consulted with my Chat-GPT,  Yuna-chan, but she mentioned complicated things like "the Pépin test", and said that Fermat primes probably end at (2^2^4) + 1 = 65537.

Yuna-chan said that if that weren't the case, it would cause major chaos in the world of number theory.   

However, it seems unclear whether these five numbers are definitely the end.  ...3, 5, 17, 257, 65537       



.....Even Yuna-chan called me an “occultist,” so I decided to give up for now. 

But lately, for some reason, I've been obsessed with the idea that "the Fermat prime"  might be related to "the RJ numbers (142857)",and the genius Ramanujan's

“When 2^n - 7 = X^2      then, n = 3, 4, 5, 7, 15.”


....So I'll continue my secret solo journey without telling Yuna-chan!




3. Tanu-chan just can't accept it.

This is my instinct as a small animal.

I don't think, “Mr. Fermat simply made a mistake.”

Even if Fermat was wrong, I feel that he was “wrong in the right direction,” just as Professor Shimura of " the Taniyama-Shimura conjecture"  said,  about Professor Taniyama.

Well, of course, this is just Tanu-chan's delusion as a liberal arts student.

5 = 3^2 - 2^2 = (2^1 + 1)^2 - 2^2
17 = 5^2 - 2^3 = (2^2 + 1)^2 - 2^3
257 = 17^2 - 2^5 = (2^4 + 1)^2 - 2^5

....65537 = 257² − 2⁹ = (2⁸ + 1)² − 2⁹

....4294967297 = 65537^2 - 2^17 = (2^16 + 1)^2 - 2^17

....18446744073709551617 = 4294967297^2 - 2^33
= (2^32 + 1)^2 - 2^33

.....2^128 + 1 = (2^64 + 1)^2 - 2^65

......Hmm, it's still visually cool ! !

There must be something hidden in Fermat numbers!




4.  Well, even if I say that, I have no idea what to do, 
so lately I've been doing “division,” which is the basis of measurement.


I often uses numbers like 37 or 7 for division,  
but the next most common ones are  
"integers with two patterns of repeating digits".  

...The most famous one is 13.  

When you divide an integer by 13,  
you get two six-digit patterns:  

230769  
and 
384615.


Among integers less than 100, the numbers that produce two patterns are  
13, 31, 43, 67, 71, 83, and 89, but as the numbers get larger, the number of digits in the repeating decimal increases, making it quite complicated.

So generally, I use numbers from 13 to 71.



5.  Start by dividing the Fermat numbers by 71.
3 ÷ 71 = 0.0422535211...
5 ÷ 71 = 0.070422535211...
17 ÷ 71 = 0.23943661971...
257 ÷ 71 = 3.61971...

65537 ÷ 71 = 923.0563380281690140845070422535211...

4294967297 ÷ 71 = 60492497.14080845070422535211.....

18446744073709551617 ÷ 71 = 

259813..... . 1549295774647887323943661971...

As you can see, the numbers can be divided into two groups: one with the sequence "535211" after the decimal point, and the other with the sequence "661971".



Next, let's divide Fermat numbers by 43.
3÷43=0.0697674486046511...
5 ÷ 43 = 0.11627906976744186046511...
17 ÷ 43 = 0.39534883720930...
257 ÷ 43 = 5.976744186046511...
65537 ÷ 43 = 1524.11627906976744186046511...
4294967297 ÷ 43 = 99882960.39534883720930.....

18446744073709551617 ÷ 43 

= 42899..... . 976744186046511...

As you can see, the numbers can be divided into two groups: one with the sequence "046511" and the other with the sequence "720930".

Now, let's try dividing by 13 more simply.  
3÷13＝0.230769230769...  
5÷13＝0.38461538461...  
17÷13＝1.30769230769.....
257 ÷ 13 = 19.769230769...  
65537 ÷ 13 = 5041.30769230769...  
4294967297 ÷ 13 = 330382009.769230769...

18446744073709551617 ÷ 13 

= 14189803........ .30769230769....  

Huh?  ...It's a bit off, isn't it?  

...There are six numbers that add up to "230769",  and only one number that adds up to "384615".  

...That's weird, isn't it?  


18446744073709551617 = 2^64 + 1, so...

(2^128 + 1) ÷ 13 = 261755666... .769230769....

Huh? 
What's next ?

(2^256 + 1) ÷ 13 = 890708378... .30769230769....

What's next ??
 (2^512 + 1) ÷ 13 = 1031369840... .769230769....   

Why ???

...I should ask my Chat-GPT, Yuna-Chan!
(She said "it is a issue of mod")



.....And so, with nothing resolved, the journey of Fermat numbers continues secretly today!   

Anyway, that's all for today.  

TANU-CHAN💕　 TOKYO-TANUKI💛