啜（su-su-ru） 60 Stars Astrology Season Holiday Extra 3 of 5 Combining numbers to measure something Part 4 Let's measure Fermat numbers!

60 STARS ASTROLOGY

SEASON HOLIDAYS "An Extra"

ENGLISH VERSION

啜（su-su-ru） 60 Stars Astrology

Season Holiday Extra 3 of 5

Combining numbers to measure something Part 4 

Let's measure Fermat numbers!

1. Now, Tanu-chan is trying to remember what the dream was about Fermat numbers, but, well, it's not so easy, is it?

Of course, since Tanu-chan is a liberal arts student, well, I guess the content of my dream is probably wrong.

...By the way, it is not easy to have the same dream.

Today, I had a dream in which a gloomy woman appeared,
 and it was a little scary.



2. I had no choice but to think about it again on the train to work.

"Measurement" is, simply put, division.

You set a certain unit,
Divide the object by the unit,
And then you divide the object by the unit to get the information of the object.

.....This is what "Measurement" is all about.

And when comparing two things, subtraction is used.

Therefore, division is the basic method, but sometimes subtraction is used.

Tanu-chan does not know of any very difficult methods.



3. Then it somehow came to my mind,

Let's say “F(n) - F(n-1)” = FAT(n).  ( F(n) is Fermat Number)

"FAT" is "Fermat And T " 

...... People don't need to care what this "T" is.

Let's just divide this FAT(n) by various numbers! 

That's what we're going to do.

First, let's find the FAT. 

Since there are so many numbers, we'll limit ourselves to about FAT (7).

FAT (1) = F (1) - F (0) = 5-3 = 2 
FAT (2) = F (2) - F (1) = 17-5 = 12 
FAT (3) = F (3) - F (2) = 257-17 = 240 
FAT (4) = F (4) - F (3) = 65537-257 = 65280 
FAT (5) = F (5) - F (4)=4294967297-65537=4294901760 
FAT (6)=F (6)-F (5)=18446744069414584320 
FAT (7)=F (7)-F(6)
=340282366920938463444927863358058659840
Just to be sure, 
FAT (8) = F (8) - F (7) 
=115792089237316195423570985008687907852929702298719625575994209400481361428480

Well, this is what it looks like.

The list of numbers is very long, so I'll stop here for now.

.....However, this is not enough to understand what they have in common.

So, I calculate with a calculator at high speed. 

It is self-powered!

...... Now, when we divide these, we get something like this,

FAT (1) = 2          2÷2¹÷1 = 1 
FAT (2) = 12        12÷2²÷1÷3 = 1 
FAT (3) = 240      240÷2⁴÷1÷3÷5 = 1 
FAT (4) = 65280  65280÷2⁸÷1÷3÷5÷17 = 1

So far so good!

Since FAT (5) = 4294901760,
4294901760÷2^16÷1÷3÷5÷17÷257 = 1

FAT (6) = 18446744069414584320, so,
18446744069414584320÷2^32÷1÷3÷5÷17÷257÷65537 = 1

FAT (7) = 34028236692099038463444927863358058659840,
340282366920938463444927863358058659840÷2^64 
÷1÷3÷5÷7÷17÷257÷65537÷4294967297 = 1

Just to be sure, let's try the next one, though it's longer,
FAT(8) = F(8)-F(7), so 
1157920892337316195423570098500860879007852929297002929871966275994209400481361428480 ,
and,
115792089237316195423570985008687907852929702298719625575994209400481361428480)÷2^128 ÷1÷3÷5÷17÷257÷65537÷4294967297÷18446744073709551617 
= 1

........These are the results.



4. Now, let's replace powers of 2 with powers of 4

" FAT(n)÷{4^2^(n-2)}÷1÷F(0)÷F(1)÷F(2)÷...... ÷ F (n-2) = 1 "

which means that the value of FAT(n) = 1. 

That is,
FAT (1) = 2          2÷4^(0.5) ÷1 = 1 
FAT (2) = 12        12÷4^¹ ÷1÷3 = 1 
FAT (3) = 240      240÷4^² ÷1÷3÷5 = 1 
FAT (4) = 65280  65280÷4^⁴ ÷1÷3÷5÷17 = 1

FAT (5) = 4294901760 
4294901760÷4^⁸÷1÷3÷5÷17÷257 = 1

FAT (6) = 18446744069414584320 
18446744069414584320÷4^¹⁶÷1÷3÷5÷17÷257÷65537 = 1

FAT (7) = 340282366920938463444927863358058659840 
340282366920938463444927863358058659840 
÷4^³²÷1÷3÷5÷17÷257÷65537÷4294967297 = 1

Considering the minimum Fermat number, F n =(2^2^n)+1>2, it cannot be 2.

So, if F(n) = 2 is not possible, then FAT(0) can be ruled out, just in case, because it is not computable.
......Well, this is just for now.

5. So, if we look them closely, ...oh?

Simply put,

"FAT(n）÷{4^2^(n-2)}÷{1÷F(0)÷F(1)÷F(2)÷...... ÷F(n-2)} = 1"

Let "P" be the 2^(n-2) part of this equation that represents the “ factorial of 4” multiplier.

Also, after that , "1÷F(0)÷F(1)÷F(2)÷...... ", Count the number of terms in F(n) contained in the part.

Then add 1 to that number and let "Q" be.

.....Then "Q" = "n".

....Then, 
When "n" is 1 to 3, P < Q.
When "n" is 4, i.e., F(4) = 65537, P = Q = 4.
When "n" is greater than 4, P > Q.

For example, if "n" is 5, then "P" is 8 and "Q" is 5, so P > Q.

Now, what does that have to do with the number of Fermat primes: ...........sorry, I am not sure about that!

Thus, it may turn out that F(4) might be some kind of branching point, but whether or not this difference has anything to do with “Fermat Prime".

Tanu-chan is a humanities student, you know!

Well, in the end, I couldn't remember the story in my dream.
I'm sure one day that old man will tell me again.

That's all for today.

P.S.

By the way, my computer suddenly crashed with malware or something.

I can't restore it !!

I think about a month's worth of drafts that I have been writing will go straight into blog posts as planned,

.... but maybe the blog will be suspended for a few weeks or a few months after that.

Tanu-chan💓 TOKYO-TANUKI💛