叕 (te-tsu) 60 Stars Astrology Season Holiday Extra 3 of 4 Combining Numbers to Measure Something Part 3 " Measuring and Fermat Numbers "

60 STARS ASTROLOGY

SEASON HOLIDAYS "An Extra"

ENGLISH VERSION

叕(te-tsu) 60 Stars Astrology

Season Holiday Extra 3 of 4

Combining Numbers to Measure Something Part 3 

" Measuring and Fermat Numbers "

1.  Well, I gave it a kind of cool title.

But, at first, well, as you all know, Tanu-chan is a liberal arts student.

So, I am not responsible for anything about this article.
....But I won't go into the hospital either.


I studied mathematics in high school, but that was about 40 years ago.

I have forgotten it.

So I am a real amateur!

By the way, I wrote a little about what I am going to write today in my Twitter (X) post.

2.  Tanu-chan sometimes has dreams and visions. ....They could be hallucinations.?

I don't see a Goddess like Ramanujan had met, I think it is an old -man somewhere.

That old man tells me many things, but I can't remember them!





3.  Fermat Numbers are the odd numbers shown by (2^2^n) + 1, right?

F(n) = (2^2^n) + 1

That's the old man that says.
When n is 0, F(0) = 2^2^⁰ + 1 = 3
When n is 1, F(1) = 2^2^¹ + 1 = 5 
When n is 2, F(2) = 2^2^² + 1 = 17 
When n is 3, F(3) = 2^2^³ + 1 = 257
and,
When n is 4, F(4) = 2^2^⁴ + 1 = 65537.

Up to this point, the numbers were prime, so Fermat said that all Fermat Numbers are prime (Fermat Primes).



However, almost 100 years later, Mr. Euler came up with
F(5)=4294967297 is 641×6700417, which is not a prime number!
and he proved it!


And after F(5), up to about F(33), until now everyone has calculated and knows that these are not prime numbers.


So then the Fermat primes are until n is 4 (65537)?

It seems that "Maniacs" are working day and night to find "the Next Fermat Prime Number".




4.  So, Tanu-chan was told something about Fermat Numbers by that old man, maybe.

Specifically, I think it was something like the Fermat number appears in the root of the solution of a quadratic equation of the form below.

Well, it's just a feeling!

First, let “n” be an integer 

and

Let us consider the equation [2^{2^ⁿ-¹-1}]X²-X-[2^{2^ⁿ-¹-1}] = 0 
.
This solution is by the formula for the solution of a quadratic equation,
X = [1±√{(1+2²×[2^{2^ⁿ-¹-1}]²}] / 2×[2^{2^ⁿ-¹-1}]


The part in this root √ that is  "1 + 2² × [2^{2ⁿ-¹-1}]²"  is the “Fake Fermat Tanuki Number”

Now, I call this "the Fake Fermat Tanuki number" as " FFT(n)".

For example, when n=1, 
FFT₁=1+2²×1²=5.

When n = 2, 
FFT₂ = 1 + 2² x (2¹)² = 17.

When n = 3, 
FFT₃ = 1 + 2² x (2^³)² = 257.

When n = 4, we have 
FFT₄ = 1 + 2² x (2⁷)² = 65537.

Additionally, if we calculate FFT₀, ....we get 3.


This is how the "FFT number" will always resemble the Fermat Number!



5.  .....And when in that dream,

"[2^{2^ⁿ-¹-1}]X²-X-[2^{2^ⁿ-¹-1}] = 0 " is, furthermore, 

transformed into,

2^{2^ⁿ-¹-1}]X²-[2^{2^ⁿ-¹-1}] = X 

[2^{2^ⁿ-¹-1}] = X / (X+1) (X-1)

and then,

[2^{2^ⁿ-¹-1}] = Φ / (Φ+1) (Φ-1) = 1

"There's a circle related to it too."

.....I think I heard him say something like that, didn't I?



.....And I think I heard something about using this equation to determine something ?? or something like that.


But I have forgotten.

I could write the formula of the above, because when I woke up in the morning, I wrote a note on a piece of paper.

Having left it for a week, I had forgotten most of the contents of the man's story.

Now, I think it was probably something to do with Fermat primes. ........???

......And so began Tanu-chan's trip down memory lane.

Though the journey was only by commuter train!




That's all for today.




Tanu-chan💓 TOKYO-TANUKKO💛