- Sep 2 3:07 AMNo Subject

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09/01/10 10:15:09

Hi

My name is François Lebrun,I have always been curious about prime numbers(PN) and i recently found that PN distribution is not random or follow a unexpected pattern, but it is systematical and can be predicted to the infinity.

The question of PN distribution may be complex but the answer is simple ! Here is my conclusion:

* 2,3,5 are accidental PN, so they should not ne considered in the search of PN distribution

* All normal numbers are multiple of 2, 3 or 5, their multiples are systematic and repetitive 2,4,6..... 3,6,9....5,10,15....Their combined distribution leave "Holes" 7,11,13,17..... I name these holes Prime Number Family( PNF). If the normal numbers are predictive and repetitive we can expect to have repetition in PNF and yes there is a repetition.

* The first eight PN( 7,11,13,17,19,23,29,31), i call them Prime Number Base(PNB) are the base of all PNF, the repetition is at 30+ all the time so 7, 37, 67.....11.41, 71.....etc

* We can express them also as : (7+30X), (11+30X),(13+30X), (17+30X), (19+30X), (23+30X), (29+30X) +(31+30X) . X can have any value from 0 to infinite and generate all the PNF. The other way around we can find if any number is a PNF with 2 simple calculations EX 119 : A- 119-19=100/30=3.333 so not an PNF B- 119-29=90/30= 3 so it is a PNF . We use -19 -29 or -7 - 17 or -11 -31 or -13 -23 depending of the end of the number tested.

*Does all PNF are PN? No , the exceptions are multiples of PNF . An example 49 is PNF 49-19/30=1 but 49 is multiple of 7X7, so a PNF but not a PN. As we know that PNF are predictive and repetitive "Holes" in the normal numbers, so all PNF that are not PN are multiples of PNF . As all PNF multiples are systematic we can find them easily and by difference all PN can be found to the infinite, let me explain.

* The eight PNB create the first group of PNF multiple that will be repeated to infinity.

7 11 13 17 19 23 29 31

7 49 77 91 119 133 161 203 217

11 77 121 143 187 209 253 319 341

13 91 143 169 221 247 299 377 403

17 119 187 221 289 323 391 493 527

19 133 209 247 323 361 437 551 589

23 161 253 299 391 437 529 667 713

29 203 319 377 493 551 667 841 899

31 217 341 403 527 589 713 899 961

* For each of these multiples we can find their PNB

Base 7 11 13 17 19 23 29 31

7 19 17 31 29 13 11 23 7

11 17 31 23 7 29 13 19 11

13 31 23 19 11 7 29 17 13

17 29 7 11 19 23 31 13 17

19 13 29 7 23 31 17 11 19

23 11 13 29 31 17 19 7 23

29 23 19 17 13 11 7 31 29

31 7 11 13 17 19 23 29 31

* And their Factors

Factor 7 11 13 17 19 23 29 31

7 1 2 2 3 4 5 6 7

11 2 3 4 6 6 8 10 11

13 2 4 5 7 8 9 12 13

17 3 6 7 9 10 12 16 17

19 4 6 8 10 11 14 18 19

23 5 8 9 12 14 17 22 23

29 6 10 12 16 18 22 27 29

31 7 11 13 17 19 23 29 31

*So for each of the Base we can find the Factors of all PNF multiple( PNFMF) (See attached Excel file)

* For example for the Base 23 the (PNFMF) are:

6 35 64 +

13 72 131 +

20 109 198 +

+ + +

4 17 30 +

15 58 101 +

26 99 172 +

+ + +

10 29 48 +

27 76 125 +

44 123 202 +

+ + +

23 54 85 +

46 107 168 +

69 160 251 +

+ + +

What we can conclude is that within the PNF with a Base 23 the following factors 4,6,10,13 are not PN while PNF with 1,2,3,5,7,8,9,11,12 are PN

1 53 PN

2 83 PN

3 113 PN

4 143 Multiple 11X13

5 173 PN

6 203 Multiple 29 X 7

7 233 PN

8 263 PN

9 293 PN

10 323 Multiple 19x17

11 353 PN

12 383 PN

13 413 Multiple 7X59

So it is simple to generate PN factors to infinity and with a simple calculation all PN to infinity.

Q Prime Numbers Formula

Prime numbers = (( {1,2,3,4,5 ........} - PNFMF7) X30) +7

( ({1,2,3,4,5 ........} - PNFMF17)X30) +17

( ({1,2,3,4,5 ........} - PNFMF11)X30) +11

(( {1,2,3,4,5 ........} - PNFMF31)X30) +31

(( {1,2,3,4,5 ........} - PNFMF13)X30) +13

(( {1,2,3,4,5 ........} - PNFMF23)X30) +23

(( {1,2,3,4,5 ........} - PNFMF19)X30) +19

(({1,2,3,4,5 ........} - PNFMF29)X30) +29

In the previous example on base 23, of 1,2,3,4,5,6,7,8,9,10,11,12,13 we have excluded the

PNFMF23 include in this series 4,6,10,13 to have 1,2,3,5,7,8,9,11,12 that are factors of PM

in the base 23. So for each of them we X 30 and add 23 Ex ((1X30)+23) = 53 a prime number

we can do the same calculation to the infinity, to find all PN.

I conclude that PN distribution is not random and can easily predicted to infinity. I have worked alone on the " Q Prime number formula" who is already deposited at a law firm. I would appreciate to receive comments or questions. You can quote this formula citing the source.

Thanks.

François Lebrun

Quebec City

Quebec Canada

Date: Wed, 01 Sep 2010 10:15:09 -0400

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