Goldbach Conjecture

all even number >=6 ，are the sum of 2 primes

All odd numbers >=9 ，are the sum three primes

I think the key word of the Goldbach's conjecture is the 3:2 the 9:6 ， the even number odd number.

Take one “stick” as one prime number, then:

The Fibonacci sequence:

The” tortoise”

                     •Three guys two girls, it is single . (3:2)

Three guys two girls, it is single ( that odd number).

It tells 3:2 hedge with 2:3. (In fact, also including 9:6)

Use the tortoise to express it (don't have to take the tortoise actually)

→ The tortoise

Red cycle is the tortoise’s M number; (odd)              Blue cycle is the tortoise’s F number (even)

Masculine number×3                                                         feminine number×2

1→3→9→7                                2→4→8→6

1×3=3     =3¹          3                 2×2=4    =2²       4 

3×3=9     =3²          9               4×2=8    =2³      8

9×3=27     =3³          7                8×2=16   =2⁴     6

27×3=81    =3⁴          1               16×2=32   =2⁵      2 

   

One again (digit of the number)                        two again (digit of the number)

81×3=243       3                                         32×2=64         4

243×3=729      9                                        64×2=128        8

729×3=2187     7                                       128×2=256       6

2187×3=6561    1                                       256×2=512       2

  

One again                                               two again

6561×3=19683       3                            512×2=1024        4   

19683×3=59049      9                            1024×2=2048       8

59049×3=177147     7                           2048×2=4096       6 

177147×3=531441    1                           4096×2=8192       2

One again                                        two again

531441×3=1594323       3            8192×2=16384        4

1594323×3=4782969     9            16384×2=32768       8

4782969×3=14348907    7           32768×2=65536        6

14348907×3=43046721   1           65536×2=1310872     2 

To plus them :

3+9+27+81=120   -------the purple number was gotten from plus them .

Red cycle is the tortoise’s M number; (odd)

9720÷120=81

787320÷9720=81

63772920÷787320=81

I always got 81

Blue cycle is the tortoise’s F number (even)

960÷60=16

15360÷960=16

245760÷15360=16

I always got 16

Now I got 81 and 16.

3⁴ =81  
 2⁴ =16

The tortoise’s Masculine number; (odd)

3=3¹ → 9=3²  → 27=3³ →    81 =3⁴ →…

Three guys : 3¹ →3²  → 3³ →   3⁴ →…odd always walk as three

The tortoise’s Feminine number (even)

              4 =2² → 8=2³   → 16=2⁴ →32=2⁵  →…

Two girls :2² →2³   →2⁴ →2⁵ →…even always walk as two

Now I see it goes back three guys’ two girls …

Odd numbers 1;3;9;7 as a group（the red）. 

Even 2;4;8;6 as a group. (the blue)

.

9：6

 9：6

Before I start the 9:6 ，let me comment something first . as follows ：

The largest number of the “tortoise” matrix is nine. (It’s an odd number)

The largest number of the “horse” matrix is ten. (It’s an even number)

If view it as base number:

The “tortoise” matrix total number is 45. 1+2+3+4+5+6+7+8+9 = 45

The “horse” matrix total number is 55.  1+2+3+4+5+6+7+8+9 +10=55

If view it as ordinal number:

The “tortoise” matrix has nine positions.

The “horse” matrix has ten positions.

  

^{The “Horse” and the “Tortoise” as square}

 ^{The “Horse’s” number is 55 and the “Tortoise’s” number is 45. Put them together is 100 (45+55=100) that is Feminine and Masculine’s full.  The full that the 100 take oblique as the boundaries to separate it ，then the 10 positions （the 10 positions means If view it as ordinal number）from the first till 10th′s it hold is 55（the 55 means If view it as base number）,}

^{if 9 positions what it hold is 45.} 

^{The two across then become the “Horse” and the “Tortoise” that two triangles shape}. ^{Whenever from the lesser position goes to more position, it must be triangle shape the base number it hold.}

^{It is triangle shape to break the 100 points’ square , (if the 100 points is square shape).  It is the following figure’s root.}

^{The “Horse” and the “Tortoise” as triangle}

 ^{In the “Horse” and the “Tortoise” as square, The “Horse” and the “Tortoise” are two numbers, but in the “Horse” and the “Tortoise” as triangle ,it is one number that two became one .}

^{the 9 layers Pascal triangle shape that the 1;2;3;4;5;6;7;8;9 I draw is the “Tortoise” (I draw it red color).  The 10 positions of the points that one two three……. until ten means the horse .(I draw it blue, the points.)}

^{In the “Horse” and the “Tortoise” as triangle, it is one number that the two become one. Although they each shape haven’t appeared yet. They position haven’t action yet .but the law of the “masculine and feminine embrace each other”.，and the law of Pascal’s triangle “shape” already stay in it silently .The following is to discuss it.}

^{10 positions of the points to response horse (horse is 10 positions of ordinal number)}

^{Three layers of blue points.}

^{Center has 9 blue points.}

^{Followed by has 2×9=18 that 18 blue points.} 

^{Outside has 3×9=27 that 27 blue points.}

^{In other words, from the inside to outside:}

^{(Center) the first layer:   9 blue points.}

^{The second layer  :        2×9=18 blue points.}

^{(Outside) the third layer  : 3×9=27 blue points.}

^{Total is 54 (9+18+27=54), except the center heart (I draw the heart purple color)} 

^{Also three layers of blue points.}

^{From the inside to outside:}

^{(Center) the first layer:  6 blue points.}

^{The second layer:        2×6=12 blue points.}

^{(Outside) the third layer: 3×6=18 blue points.}

^{Total is 36, (6+12+18 = 36) except the center heart (I draw the heart purple color)}

 Also three layers of triangle.

^{From the inside to outside:}

^{(Center) the first layer:  nine triangles.}

^{The second layer  :    3×9=27 triangles.}

^{(Outside) the third layer  : 5×9=45 triangles.}

^{Total is 81.}

 Three layers of triangle:

^{(Center) the first layer:    6 triangles.}

^{The second layer  :       3×6=18  triangles.}

^{(Outside) the third layer  :  5×6=30 triangles.}

^{Total is 54. (6+18+30 = 54)}

Although the products of the number are different in the above each pictures. The essence of the above each pic is the same by the “nine” and the “six” of it that it’s one same thing.

The nine and six could have three layers from center

^{(Center) the first layer→ The second layer → (Outside) the third layer}

On every layer to discuss it ：

Nine is the male relatively here, the masculine surround feminine. Six is the female relatively here, the feminine including masculine.

My picture comment of the fractal of the nine and six:

 The “masculine and feminine Embrace each other”.

My picture comment of the “Horse” and the “Tortoise” is one number.

Or：

My picture comment of the “Horse” and the “Tortoise” are two numbers.

For example: the Koch snowflake

http://en.wikipedia.org/wiki/Koch_snowflake

I think if the Koch snowflake want to grow big ,everytime of the Koch snowflake grow up is about the 9:6 of the nine‘s shape and the six ‘s shape in it.

 other  9：6

● 1 3 7 9

■ 2 4 6 8

 According to odd number is● round shape, even number is ■shape.

6 should be ■, 9 should be ●.

But the six also could be ●shape congenitally，and the nine also could be ■shape congenitally.

So ━ The 9:6 . 


why its 6 around one ，not 4  around one or 5 around one ？

“Round of seven” The center of the “Round of seven” is six surround one. They edge touch each other don’t have gap.

What is the “don’t have gap”? Only the “6 surround one” don’t have gap, When all the rounds their diameter is equal.

For example：

Let all the rounds their diameter is equal .the “five surround one”has gap.

 Let the rounds their diameter is equal. The “four surround one “has gap.

Why ● means odd number？

                    why■  means even number？

When a round’s diameter is equal with a square’s side length.

 The Square with the round has the relationship of masculine and feminine.

If let the round is M, then the square is F.

 If let the square is M then the round is F.

 when a round’s diameter is equal with a square’s side length. The ●is guy (masculine)，the ■ is girl(feminine). As if they let odd number is masculine then even number would be feminine .

^{Masculine across is one “big stick” relatively, while feminine across is two “little stick” relatively} 

^{Even number could be the “a pair of numbers” congenitally and relatively .for example ：Eight is a pair of four. Four is a pair of two. We could get the “single” from “half” the even number.}

Prime number could be a single number congenitally and relatively; Irrational number could be a single number congenitally and relatively.

  

^{Actually, any number could be the single number. It relates to distinguish masculine and feminine. The Number’s relative motion is not static or the masculine and the feminine won’t relatively static. So, it depend the specific situation; specific case or the math player’s desire. The key word of it is “half it or double it”. If “double” the single number then could get the “a pair of numbers”. If “half” the “a pair of numbers” then could get the single number.}

When a round its diameter is one ，its perimeter isπthat 3.1415926……

πthat 3.1415926……is kind of single number.

When a square side length is one ，its perimeter is four.

According to odd number is guy(masculine) ，even number is girl (feminine) they set .

So，●means odd number.

■means even number.

This ideal may also is that ：

1 ÷ 1 and 1×1= one still；  So one don't join the “start” .

Odd start from three. Even start from two in the Multiplication and division.

Too extreme as if zero in the addition and subtraction, too extreme as if one in the Multiplication and division

 I notice a situation. When the text book try to express the “Pascal’s Triangle shape” I called in English. It would say “Guy one, girl two; Guy three, girl four; Guy five, girl six; Guy seven, girl eight; Guy nine, girl ten” that guy or girl in the front .number in the back.
But when it tries to express 3:2 .it won’t put guy or girl in the front.It would say “three guys two girls, it is single”. that Guy or girl in the back.

What I did:

Now the “three guys two girls, it is single”,  Guy or girl in the back.

 According the semantic meaning of the ancient original of “Three guys two girls, it is single 〞.I also could call it: “three guys two girls, it is sharp shape” in English.

According ancient Chinese guys think or Benoit Mandelbrot’s to the triangle.

I think the “three guys two girls, it is single” could be irrational number ； Also could be natural number.

In the Fibonacci sequence：

（1）1

 　　（2）1

 　　（3）2（1+1）

 　　（4）3（1+2）

 　　（5）5（2+3）

 　　（6）8（3+5）

 　　（7）13（5+8）

 　　（8）21（8+13）

 　　（9）34（13+21）

 　　（10）55（21+34）

 　　（11）89（34+55）

 　　（12）144（55+89）

 　　（13）233（89+144）

 　　（14）377（144+233）

 　　（15）610（233+377）

 　　（16）987（377+610）

 　　（17）1597（610+987）

 　　（18）2584（987+1597）

 　　（19）4181（1597+2584）

 　　（20）6765（2584+4181）

 　　（21）10946（4181+6765）

 　　（22）17711（6765+10946）

 　　（23）28657（10946+17711）

 　　（24）46368（17711+28657）（46368÷28657=1.61803398820......）

 　　......

With the increase in the number series, the ratio of the previous one and after a closer and closer to the golden section value 0.6180339887 ...

3 ÷ 2 = 1.5,

5 ÷ 3 = 1.666 ...,

8 ÷ 5 = 1.6

89 ÷ 55 = 1.6181818 .

233 ÷ 144 = 1.618055 ...

75025 ÷ 46 368 = 1.6180339889 .... ..

the More goes to the back, the ratio is more closer to the golden ratio.

Start from 3：2（ In the Fibonacci sequence）

And according to the Fibonacci sequence appear 5 actions：

3+2=5 ； 21+34=55 as if 3+2=5 ； 233+377=610 as if 3+2=5  ………）

I think ：One case of the “three guys two girls, it is single” is that When western people say the golden ratio would feel like the “three guys two girls, it is single”.

If let the 0.618 is Masculine， then the 1.618 is feminine.

If let the 1.618 is Masculine， then the 0.618 is feminine. 

another 9:6

If  consider the situation of “M.I.F.R” and “F.I.M.R” ， As follows:

If I don't consider the situation of the “M.I.F.R” and “F.I.M.R”  now ，but consider the hedge number to arrange the order .

Odd number is   1→3→5→7→9

 Even number is    4→2→10→8→6

『one ,nine』    1+9=10     They are 《hedge number》

『two ,eight 』 2+8=10      They are 《hedge number》

『three, seven』 3+7=10    They are 《hedge number》

『four ,six 』    4+6=10    They are 《hedge number》

Meanwhile

【 one with six】     6-1=5        1-5=-5 

           【 seven with two】  7-2=5        2-7=-5

          【 three with eight】 8-3=5        3-8=-5

          【 four with nine】   9-4=5        4-9=-5

                 they are 《same root number》.

Odd number 1→3→5→7→9  to ×1；2；3；4；6；7；8；9

Even number 4→2→10→8→6 to × 1；2；3；4；6；7；8；9

Write the single-digit only：

Observe it (single-digit only)：

 First I Observe the Odd number 1→3→5→7→9 ：

Whenever hedge number it would mirror symmetry

The Case:

1 with 9 is hedge number

It is mirror symmetry

1       3  5  7  9┊9  7  5   3   1

2 with 8 is hedge number

It is mirror symmetry

1    6  0  4  8┊8  4  0  6  2

3 with 7 is hedge number

It is mirror symmetry

3  9  5  1  7┊ 7  1  5  9  3

4 with 6 is hedge number

It is mirror symmetry

4 2 0 8 6 ┊6 8 0 2 4

Whenever same root number, it would same root!

the Case:

1 with 6 is same root number

【 one with six】     6-1=5        1-5=-5 

【 seven with two】  7-2=5        2-7=-5

【 three with eight】 8-3=5        3-8=-5

【 four with nine】   9-4=5        4-9=-5

they are 《same root number》.

2 with 7 is same root number

【 one with six】 【 seven with two】 【 three with eight】 【 four with nine】    they are 《same root number》.

Other case of the “Whenever same root number, it would same root” is the same with that .（so I omit it ）.

Now I observe Even number  4→2→10→8→6

Whenever hedge number it would mirror symmetry

4 2 0 8 6┊6 8 0 2 4

Other case of “Whenever hedge number  it would mirror symmetry “of Even number  4→2→10→8→6 （I omit it）

Whenever it’s same root number, it would be the same

Let the Odd number 1→3→5→7→9 is M， then the Even number  4→2→10→8→6 would be F.

It  appear 9:6

only When 1 and 4 ；6 and 9 could appear the order that odd number is 1→3→5→7→9 .   Even number is 4→2→10→8→6

As follows

The 6 and 9 

New born number             Mature number

only Mature number have the qualifications to tell 9:6.

cuz New born number only have 4 positon,have not become Decimal yet.

Figure of when 13579 go with 421086

1→3→5→7→9 could go with 4→2→10→8→6

Figure of why 1→3→5→7→9  could go with  4→2→10→8→6

Explain why take nine and ten:

  9 is the largest odd number.  10 is the largest even number (in the 1~10). The largest number of the “tortoise” matrix is nine. The largest number of the “horse” matrix is ten. The “tortoise” matrix has nine positions. The “horse” matrix has ten positions. The Nine represent the “tortoise” matrix. The Ten represent the “horse” matrix.

The “tortoise” self multiplies self:

nine positions self multiplies self that 9×9=81

Ancient Chinese always says “double nine would go back one”.

The “horse” self multiplies self:

Ten positions self multiplies self that 10×10=100

Odd number start from one ，So only take one piece .                  

even number start from four，   so take four sides.

The “Horse” matrix appear the 1→3→5→7→9 could go with 4→2→10→8→6

The red circle that masculine number : 1→3→5→7→9 is a Möbius strip.

The blue circle that feminine number : 4→2→10→8→6 is a Möbius strip.

The two Möbius strips that  1→3→5→7→9  and  4→2→10→8→6 could become a Klein bottle.

The feminine number : 4→2→10→8→6 start from four end in six . Six is the super feminine here. So, six representative feminine number.

The masculine number : 1→3→5→7→9 start from one end in nine. Nine is the super masculine here. So, nine representative masculine number.

④ Across appear the 9:6  

3×3=9   or    3+3+3=3=9(Romeo)        6×3=18   or  6+6+6=18(Romeo)

3×2=6   or    3+3=6(Juliet)              6×2=12   or   6+6=12(Juliet)    

           9:6                                    18:12 →9:6

  9:6