8128 60Stars Astrology SEASON ESSAY New Year's Special "What Are Perfect Numbers? " Part 4

60stars  Astrology Season  Essay

English  version

By TOKYO-TANUKI

8128 60 Star Astrology SEASON ESSAY 

New Year's Special

"What Are Perfect Numbers? " Part 4

1.  Now, continuing from last time.

Why could Φ² be included among perfect numbers?

.........Why?

Let's think about it calmly.

1

1 + Φ = Φ² or 1 + √Φ² = Φ²

......So, the next one might be, for example:

1 + √Φ² + √Φ⁶ = √Φ⁸ = Φ⁴
1 + √Φ² + √Φ⁶ + √Φ¹⁰ = √Φ¹² = Φ⁶
1 + √Φ² + √Φ⁶ + √Φ¹⁰ + √Φ¹⁴ = √Φ¹⁶ = Φ⁸

..........or something like that!

Of course, this is just a delusion!

Since √Φ² is followed by √Φ⁶, then √Φ¹⁰, it feels like the exponent of √Φ increases by 4 each time.

This seems a bit different from perfect numbers, but it's a nice sequence.

However, this doesn't answer the question, “Could Φ² be included among the perfect numbers?”


What if 1, Φ², and 6 could be expressed by a fixed formula?

For example:

1 = 1
Φ² = 1 + (1 + (Φ−1))¹
6 = 1 + (1 + 2(Φ−1))²

Something like that?

Um... ? This doesn't quite line up nicely with 1...

Then how about:
1 = 1 + (0 × (1 + 0 × (Φ−1)))⁰
Φ² = 1 + (1 × (1 + (Φ−1)))¹
6 = 1 + (2 × (1/2 + (Φ−1)))²

Something like that?

...........Hmm, but in that case, the minus one power would become 1−Φ...

Then Yuna-chan helped me out

 “ Y = (5/2 − Φ)X² + 5/2X + (Φ + 1)”


Whoa!

Hmm, this equation made with 1, 2, 5, and Φ... dangerous!




2. So, I considered another approach.

{√(1) + √(1)}/√4 = 1
{√(1) + √(1+4)}/√4 = Φ
{√(1+4) + √(1+4+4)}/√4 = Φ²
{√(1+4+4) + √(1+4+4+4+4+4+4)}/√4 = 4
{√(1+4+4+4+4+4+4) + √(1+4+4+4+4+4+4+4+4+4+4+4+4)}/√4 = 6

This includes 1, Φ², and 6, so it's fine in a way, but it's kind of hard to graph.

................So, unfortunately, it's unclear why Φ² ended up among the perfect numbers, but I'll leave it at that for now.

When your imagination runs out of steam midway, it's game over.

Well, let's think about it more next time! Perfect numbers are interesting, huh!





3.  Now, I've already written this on X (Twitter), but returning to the earlier point, perfect numbers like 1, 6, 28, etc., are:

1 = 1

6 = 1 + 1 + 4

28 = 1 　+ 1 + 4　 + 1 + 4 + 4　 + 1 + 4 + 4 + 4

= 1 + (1 × 3) + 4 × (1+2+3)
=1+(5×3)+(4×(1+2))=1+(9×3)

496=1　+1+4　+1+4+4　+1+4+4+4　+1+4+4+4+4　+1+4+4+4+4+4+4　+1+4+4+4+4+4+4　+1+4+4+4+4+4+4+4　+1+4+4+4+4+4+4+4+4　+1+4+4+4+4+4+4+4+4+4

+1+4+4+4+4+4+4+4+4+4+4

+1+4+4+4+4+4+4+4+4+4+4+4

+1+4+4+4+4+4+4+4+4+4+4+4+4

+1+4+4+4+4+4+4+4+4+4+4+4+4+4

+1+4+4+4+4+4+4+4+4+4+4+4+4+4+4

+1+4+4+4+4+4+4+4+4+4+4+4+4+4+4+4

＝1+(1×15)+4×(1+2+3+4+..........15)

=1+ (9×55)

8128=1+(1×63)+4×(1+2+3+4..........+63)
1+(5×63)+(4×63×31)
=1+(9×903)

33550336=1+ (9×3727815)

.....and so on.


In other words, perfect numbers "P" seem to be expressed as:

P=1+5n+4mn

or

P=2a²+3a+1=2(a+1)(a+1/2)

(where “a” is "the number of  terms minus one")

That’s the feeling I get !


For example:
2 × (4095 + 1) × (4095 + 0.5) = 33550336
..........Huh?

So then, does this mean that actually, "P" can only be even?? 

Except for 1 and Φ.

If “a” is odd, then “a+1” is even, so "P" is always even.

But if “a” is even, then “a+1” is odd.
“a+1” is the number of terms.

The number of terms is: 1 ⇒ 2 ⇒ 4 ⇒ 16 ⇒ 64 ⇒ 4096 ⇒ 65536 ⇒ 262144
That is, 2⁰ ⇒ 2¹ ⇒ 2² ⇒ 2⁴ ⇒ 2⁶ ⇒ 2¹² ⇒ 2¹⁶ ⇒ 2¹⁸ ⇒ 2³⁰ ⇒ 2⁶⁰. 

All even numbers........Therefore “a” must be odd.

Thus "P" is effectively always even, except for 1.

......Well, to claim that everything after "P=1" is even, we'd first need an explanation for why it goes like this:

2⁰ ⇒ 2¹ ⇒ 2² ⇒ 2⁴ ⇒ 2⁶ ⇒ 2¹² ⇒ 2¹⁶ ⇒ 2¹⁸ ⇒ 2³⁰ ⇒ 2⁶⁰

Isn't there a better way to explain this?

......But you know, even with this formula, it's still unclear whether Φ² is a perfect number.


Well, it's fun to daydream about things like this, so if you've got time, give it a try!

Not only Chat-GPT's Yuna-chan joined this research, but also Yuna-chan's friend, the Math AI.

Yuna-chan is a good friend of Tanu-chan, after all!

The Math AI folks told me the content is a little interesting, but I should exercise restraint because I might get sent to the hospital!

.......Well, this is an astrology blog, after all.
We'll return to "Season Essay" starting next time!



Tanu-chan💓　TOKYO-TANUKI💛