Φ²¹ 60stars Astrology Season Essay Φ²¹ "The true companions of Φ"

60stars  Astrology Season  Essay

English  version

By TOKYO-TANUKI

Φ²¹　60stars  Astrology Season  Essay　Φ²¹　"The true companions of Φ"

1．Now, regarding the relationship between Φ and the Fermat numbers—at its root, this is another story of what Tanuki used to call “Φ’s search for friends.”
Depending on one’s perspective,
Φ’s friends could be 2 + √3, or (√3 + √7) / 2, or (1 + √13) / 2—
there are many possibilities.

But among them, those that behave similarly
when inverted and negated—those whose pattern resembles the Fusion–Generation formula

"QXⁿ ± R(1 / Xⁿ) = P"

—these are clearly the true companions of Φ.

2. First, of course,Φ

Φ − (1 / Φ) = 1.
Everyone knows this one.
The difference between the number and its reciprocal is 1.

1.618033988... and 0.618033988...The digits after the decimal align beautifully: 618033988...

Next, √2 + 1 = 2.41421356...
Its reciprocal, 1 / (√2 + 1) = 0.41421356...
The difference is 2.
Again, the digits line up nicely: 41421356...

Then, (√17 + 1) / 4 = 1.28776406...
The reciprocal is 0.78776406...
Not bad—most of the digits after the decimal match:776406...
The difference is 0.5 = 1/2.

Now, jumping ahead a bit:
(√257 + 1) / 16 = 1.06445122...

The reciprocal is 0.9345122...

The difference is 0.125,and the decimals more or less align again: 45122......That difference, 0.125 = 1/8 = 1 / 2³.

Next—oh? The digits don’t seem to align at first...
Let’s see:

{(√65537) + 1} / 256 = 1.00391387936542764158...
Its reciprocal: 0.99610137936542764158...

Ah! From the middle—7936452764158...—they align again.
Good!
The difference is 0.0078125 = 1/128 = 1 / 2⁷.

And next, what about
{(√4294967297) + 1} / 65536 = 1.0000152589054778218201585508...
The reciprocal is
0.9999847413273528218201585508...

Hmm, from the middle—8218201585508...—they match again.
The difference is 0.000030517578125 = 1 / 32768 = 1 / 2¹⁵.



....Wait—what’s next?
(√18467440... + 1) / 4294967296 = 1.0000000002328306...
Subtracting its reciprocal, the difference is 0.0000000004656612873077392578125 = 1 / 2³¹.



....And you see, the journey of the Fermat numbers
turns out to be the same as the journey of Φ’s friends.

The differences between the numbers and their reciprocals follow the pattern:

1 / 2⁻¹, 1 / 2⁰, 1 / 2¹, 1 / 2³, 1 / 2⁷, 1 / 2¹⁵, 1 / 2³¹ ...

In other words, Φ’s companions are those whose difference from their reciprocals takes the form 1 / (2^(2ⁿ − 1)).

3. Wait a moment!

"Could there even be a case of 1 / 2⁻²?"

"Hold on—weren’t the Fermat numbers 3, 5, 17, 257...?　That seems... a little off, doesn’t it? "　

"Is （1／√２＋１）correct ?... it seems to be（1／√３＋１） ？"

 

"It looks like Mersenne Numbers......?"

.........Yes—exactly.

That’s the perspective that matters.

Where, and how, do the Fermat numbers and Φ intersect?

.........What makes Fermat so great is that he saw that crossing point—and recognized it as something profound enough to write down.

Even if the Fermat numbers themselves are not all prime, that’s fine.

............What matters is that he caught a glimpse of truth in a single instant—and left it for us to find.

.....Truly, Master Fermat!

（to be continued)

Tanu-chan💓　TOKYO-TANUKI💛