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Distribution of prime numbers

Moussa Ouannas (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 :

 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 19n-20b 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 19n-20b 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 19n-20b 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 19n-20b 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 19(1+6a) +18k 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 19(5+6a) +18k 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 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 np N np n np 1 2 2 3 4 5 6 7 +7k 8 7 11 10 6 5 9 4 2 3 4 6 6 8 10 11 +11k 7 11 9 4 10 6 8 5 2 4 5 7 8 9 12 13 +13k 11 9 8 5 4 10 7 6 3 6 7 9 10 12 16 17 +17k 10 4 5 8 9 11 6 7 4 6 8 10 11 14 18 19 +19k 6 10 4 9 11 7 5 8 5 8 9 12 14 16 22 23 +23k 5 6 10 11 7 8 4 9 6 10 12 16 18 22 27 29 +29k 9 8 7 6 5 4 11 10 7 11 13 17 19 23 29 31 +31k 4 5 6 7 8 9 10 11

 8 9 9 10 11 12 13 14 +7k 8 7 11 10 6 5 9 4 13 14 15 17 17 19 21 22 +11k 7 11 9 4 10 6 8 5 15 17 18 20 21 22 25 26 +13k 11 9 8 5 4 10 7 6 20 23 24 26 27 29 +17k 10 4 5 8 9 11 23 25 27 29 +19k 6 10 4 9 28 31 +23k 5 6

 15 16 16 17 18 19 +7k 8 7 11 10 6 5 24 25 26 28 28 30 +11k 7 11 9 28 30 31 +13k 11 9 8

 22 23 23 24 25 26 27 28 +7k 8 7 11 10 6 5 9 4

 29 30 30 31 +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 Y3 Y4 Y5 8 6 11 4 Y2 Y3 11 5 Y2 Y3 Y4 Y5 8 4 11 6 Y2    Y3 Y4 Y5 8 9 11 7 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.

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 6410 times.