Prime numbers. Discovering symmetries between prime numbers. The Center Number. Part 4.

Source code of this part can be found at Github.

Part 1., Part 2. and Part 3. was about some numerology based equations.

sin(101) = 0.98162718344766397571
cos(101) = -0.19080899537654480436
tan(101) = -5.1445540159703107008
sin(103) = 0.97437006478523524589
cos(103) = -0.22495105434386480914
tan(103) = -4.3314758742841590333

I cannot explain why the main requirement is take 3 numbers if angle is build of 3 numbers.
As you can quickly calculate with 2 or 4 taken numbers it does not work in a symmetry context.

After this discovery I get an idea to test symmetry of known prime numbers. I asked myself:

Is there a center number between two primary numbers?

Of course there is.

I had discover, that every prime number has a “twin” at the same length from center number.

If you check https://banit.pl/wp-content/uploads/2019/08/result.txt you will find:

Size:	2		
Count:	24		
---------
Size:	6		
Count:	46		
---------
Size:	10		
Count:	32
...
  • Size is length between 2 prime numbers abs(prime1 - prime2) = size
  • In range <0, 500>:
    • there are 24 * 2 prime numbers so there is 24 symmetries
    • there are 46 * 2 prime numbers so there is 46 symmetries

And so on.
Next files are more detailed.

In https://banit.pl/wp-content/uploads/2019/08/result_details.txt you can confirm:

Size:	2		
Count:	24		
___DETAILS___
Below:	3
Center:	4
Upper:	5
---
Below:	5
Center:	6
Upper:	7
---
Below:	11
Center:	12
Upper:	13
---
Below:	17
Center:	18
Upper:	19
---
Below:	29
Center:	30
Upper:	31
---
Below:	41
Center:	42
Upper:	43
...

...
Size:	6		
Count:	46		
___DETAILS___
Below:	5
Center:	8
Upper:	11
---
Below:	7
Center:	10
Upper:	13
---
Below:	11
Center:	14
Upper:	17
---
Below:	13
Center:	16
Upper:	19
---
Below:	17
Center:	20
Upper:	23
---
Below:	23
Center:	26
Upper:	29
...

And you can define:

UpperPrime – LowerPrime = Size
and Size is given by 4n + 2 formula

File https://banit.pl/wp-content/uploads/2019/08/result_number_symmetry.txt confirms previous. File is grouped by number and it shows where the prime number exists in symmetry:

NUMBER:	3			
	C	4		P	5		L	1
	C	8		P	13		L	5
	C	10		P	17		L	7
	C	16		P	29		L	13			
	C	20		P	37		L	17			
	C	22		P	41		L	19
...

...
NUMBER:	5			
	C	4		P	3		L	1			
	C	6		P	7		L	1			
	C	8		P	11		L	3			
	C	12		P	19		L	7			
	C	14		P	23		L	9			
	C	18		P	31		L	13			
	C	24		P	43		L	19			
	C	26		P	47		L	21

C: center number
P: prime in pair
L: length in center number

So:

  • number 3 (confirmed in NUMBER: 5)
    • has a 5 prime number twin
    • symmetry size is equal 2: 5 - 3 = 2
    • center number is 4: (5 + 3) / 2 = 4
    • length from center number is equal 1: 3 + 1 = 5 - 1
  • number 5
    • has a 47 prime number twin
    • symmetry size is equal 42: 47 - 5 = 42
    • center number is 26: (47 + 5) / 2 = 26
    • length from center number is equal 21: 5 + 21 = 47 - 21

Part 5. of Prime Numbers topic is available.

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