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Ouannas, M. (2011). Distribution of prime numbers. PHILICA.COM Article number 216.

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Distribution of prime numbers

Moussa Ouannasunconfirmed user (Bejaia University)

Published in matho.philica.com

Abstract
In this paper i present the distribution of prime numbers which was treated in many researches by studying the function of Riemann ; because it has a remarkable property ; its points of discontinuity are prime numbers ; but in this work i will show that we can find the ditribution of prime numbers on remaining in natural numbers integers only.

Article body

 

Introduction :     

 The bone Ishango discovered there are nearly 20000 years and yet, it seems that we are only at starting point of understanding the prime numbers, we ignore how they are distributed, how they behave among themselves or in set of natural numbers.

 

However, you will see in this studying that there is a mean to clarify the distribution on using natural numbers only.

I think all the formulas and technics to run anything in this world already exist but only hidden, simply wait for the « moment » to remove the membrane that blind us according to some caractéristics which only God know them .

 

Spliting up of the natural integers :

 

I established a table where the first line start from 1 until  19 and the second from 20 to 38 ….so forth  as in Fig I.

 

After coloring the prime numbers in blue  and the non-prime in red ; it appears that they are adjusted (from left to right and from top to bottom) according to two (02)  oblique lines of the following formulas :

 

 

 

 

 

 

 

·        19(5+6a) + 18 k              / n = 5+6a  et n = 1+6a avec n et a € N

·        19(1+6a) + 18k                k € N       0≤ k ≤18

 

 

It appears also that they are adjusted (from left to right and from bottom to top) according to four (04) oblique lines  of the following formula :

 

*  19n – 20k      / n and k € N  0≤ k ≤18  with n ≠5b / b € N

 

 

I noticed  that the two (02) lines and the four (04) intersect in 08 points so we obtain  groups of 08 numbers.

But the 08 numbers are not all prime numbers ; those which are non-prime are a product of two prime numbers from 7 to infinity or the power of some prime numbers, «  8 is the rank of 19 in set of prime numbers » ; is it here a coïncidence ?

 

It appears clearly that the prime numbers are distributed in groups of 08 numbers which can be obtained with the following formula :

 

         2x3x5 n +2k+3  / n  an k € N   and k= {2, 4, 5, 7,8, 10, 13,14}

 

Thus , the numbers 2k+3 which  are 7,11,13,17,19,23,29,31 constitute the basis  not only for  the obtention of the prime numbers but to determine their rank(rate) and also for eliminating the non-prime numbers.

 


 FIG 1 :

 

932

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19n-20b

 

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19(1+6a) +18k

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1

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prime number

 

 

non prime number

 

 

2x3x5 n

 

 

 

 

 

 

 

 

 

 

 

2, 3, 5 those prime numbers did not belong to the distribution, but are a part of the formula, and the group of the 08 numbers  start with the number 7 in the rank 4 as represented in FIG 2.

 

 

FIG 2 :                                                                       4    30(n-1)+7

                                  30(n-2)+17            7                        

                                        30(n-1)+29          10               8   30(n-1)+19

                                                                               30n

                                          30(n-1)+11             5                 11 30n+31

                                                                                                    

                                                    30n+23              9                  6 30(n+1)+13

                                                                                                      

 

I replaced 7, 11,13,17,19,23,29,31 by their ranks   4,5,6,7,8,9,10,11; because it is a mean to determine the rank of the prime numbers.

I also established a table of non-prime numbers periods which belongs to the distribution; those numbers are the basis of primality testing.

 

FIG 3: Table of: - Primality testing

- Periods of non-prime numbers (np)

 

n

np

N

np

n

np

n

np

n

np

n

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N

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n

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+7k

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+17k

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+29k

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+31k

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+11k

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+13k

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+17k

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+19k

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31

 

 

 

+23k

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15

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+7k

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24

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+11k

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+13k

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+7k

8

7

11

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6

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29

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+7k

8

7

11

10

 

 

The periods of the power of 2x3x5n+2k+3=X+2k+3=Y

X+7

X+11

X+13

X+17

X+19

X+23

X+29

X+31

Y2

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8

6

11

4

Y2

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11

5

Y2

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Y5

8

4

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6

Y2   

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8

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Y2   

Y3

 

 

11

8

Y2   

Y3

Y4

Y5

8

7

11

9

Y2   

Y3

11

10

Yn  

11

   

 

 

 

 

 

 

TEST OF PRIMALITY:

  We choose X a natural number, if its last digit number is 7, 9, 1 or 3 it may be prime, so to know if it belongs to the group of 8 , we must do a subtraction:

If its digit number is 7 we retrench 7 or 17:

 a/ if X-7  is not  divisible by 2x3x5 i.e. 30, X is not prime and if it is divisible we obtain n; since X belongs to the group  of 8; so we verify if it is not non-prime in the table on writing n in the forms:

n = α+ (2k+3) β   according to the table of FIG 3

If it is so, X is a prime number.

b/ if X is verified with 7 we do not need verify with 17, if not we make the same process with 17.

Thus:

-         if the last digit number of X is 9 we retrench 29 or 19

-         if the last digit number of X is 1 we retrench 31 or 11

-         if the last digit number of X is 3 we retrench 23 or 13

and we proceed as previously.

 

Examples:

X=917

* X-7= 910/30 =30,333 not divisible by 30, it does not belong to the distribution it is not a prime number.

* X-17 = 900/30 = 30 =n divisible by 30, so X belongs to the distribution and its position is P7, now we give the forms of n=30:

n=2+7k; n=8+11k; n=4+13k; n=13+17k; n=11+19k; n=7+23k; n=1+29k

Now: for n=2+7k np in P7 and in P11             

      n=11+19k→ np in P11

                n= 4+13k→ np in P9

                n= 30+7k , 11k , 13k, 19k,23k,29k,31k →np in P7,P9,P11 and P6

 

Since X is in the position P7 so it is not prime.

 

NB:

 

We can also form prime numbers as great as possible   

Examples:

 

We choose n = 1600, so the eight (8) numbers are:

 

In

·        P4 → 2x3x5x1600+7 = 48007

·        P5 → 2x3x5x1600+11 = 48011

·        P6 → 2x3x5x1600+13 = 48013

·        P7 → 2x3x5x1600+17 = 48017

·        P8 → 2x3x5x1600+19 = 48019

·        P9 → 2x3x5x1600+23 = 48023

·        P10 → 2x3x5x1600+29 = 48029

·        P11 → 2x3x5x1600+31 = 48031

Now we give the forms of n:

1600=4+7k →np in P6                 n=4+19k→np in P6

1600=5+11k                                 n=13+23k→np in P6

1600=9+43k→np in P11               n=166+239k→np in P5

1600=2+17k                                  n=19+31k→np in P8/1600=14+61k→np in P4

So numbers in P4, P5, P6, P8, P11 are non-prime, only numbers in P9, P10, P7

48023, 48029 and 48017 are primes.

 

DERMINATION OF THE RANK:

 

The last digit of any prime number is 7, 9, 1 or 3, so on verifying its primality, if we retrench:

-         7  its position will be in the same position as 4

-         17 its position will be in the same position as 7

-         19 its position will be in the same position  as 8

-         29 its position will be in the same position as 10

-         11 its position will be in the same position as 5

-         31 its position will be in the same position as 11

-         13 its position will be in the same position as  6

-         23 its position will be in the same position as 9

 

The rank will be:

 

Rn = 8 n + Pk - np 

    * Rn    is the rank

    * Pk     is the position of the prime number with {4, 6, 7, 8, 9, 10, and 11}                   

    * np    the number of the non-prime numbers belong to the distribution before         this prime number 

 

FIG4:

                                                                              P  4    

                                                              P  7                        

                                                                   P 10             P  8  

                                                                               30n

                                                                          P 5               P  11

                                                                                                    

                                                                                 P 9                 P 6

 

                                          

    


I tried until 50000 in order to find a formula of the quantity (the number) of non-prime numbers and I found that it is in relation with the table of primality testing

 

 

CONCLUSION:

 

As a conclusion to this studying, i think that i clarify the distribution of prime numbers and find a mean to calculate them by splitting up the natural integers on using 19 and we can obtain this result only with 19 and 11.

 

On the other hand, it is not necessary to look for the density of prime numbers since we can talk about ranks.

 

In fact I have not found this distribution, but just remove the membrane which blinds us.

 

In addition to that, I want to tell that I have found other mysterious secrets of prime numbers but which I cannot disclose now.  

 

 

 

 





Information about this Article
Peer-review ratings (from 1 review, where a score of 100 represents the ‘average’ level):
Originality = 306.25, importance = 25.00, overall quality = 6.25
This Article was published on 27th January, 2011 at 11:31:34 and has been viewed 6175 times.

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The full citation for this Article is:
Ouannas, M. (2011). Distribution of prime numbers. PHILICA.COM Article number 216.


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1 Peer review [reviewer #10906confirmed user] added 22nd October, 2012 at 03:34:01

The article does not define all its terms. “Retrench” seems to mean “subtract”, but the author also writes “Subtraction” at one point so this does not seem to be a language problem. The next unknown word was “period”. This could have had several meanings, and I’m sure I could eventually have puzzled out which was intended by pouring over the examples.

The basic idea is to deal with small primes by looking at them modulo 30. This is not clarified by the author’s insistence on making tables modulo 19! While 19 is the 8th prime, the number 8 appears because there are 8 numbers modulo 30 which are coprime to 30.

Originality: 7, Importance: 2, Overall quality: 1




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