2 ABSTRACT To this day, however, there is a common view bordering on the certainty that prime numbers are arranged chaotically on the number line - they are not governed by any laws that would allow us to describe them all. This view is argued by this article "Spirals of prime numbers" and tries to show what laws govern the spiral sequences of prime numbers and their products greater than 3, their mutual dependence, how they complement each other and how, based on these laws, all prime numbers and their products can be describe. SUCCESSION OF PRIMARY NUMBERS AND THEIR PRODUCTS Succession of prime numbers and their products creates the arithmetic sequence 2-3-5-7-9-11-13-1517-19-21-23-25-27-29-31-33-35 -..., which each word (starting from the third) is formed by adding a constant number “d” to the previous term, called arithmetic sequence difference (3 -2-5 -2- 7 -2- 9 2-11 ..).
Interestingly, the products of the number 3 (9 -6- 15 -6- 21 -6-27) create their own sequence with a constant difference of 6 and this fact makes this sequence a convolution of three sequences; products of 3, and two sequences of prime numbers and their products greater than 3 (5 -6- 11 -6- 17 -6- 23 -6- 29 -6- 35, ..) (7 -6- 13 -6- 19 -6-25,) with constant difference d = 6. The products of the number 3 by their constant spacing 6 in which they follow have a decisive influence on the number of primes and their products greater than 3. If up to 100 from 9 to 99 there are 16 products of the number 3 (100 - 4)/6 = 96/6 = 16 , then prime numbers and their products greater than 3 can only be 50 - 16 = 34. There are 6 products of the number 5 (25 -30- 55 -30- 85, 35 -
3 30- 65 -30- 95), and 3 products of the number 7 (49 -28- 77 -14-91), that is, subtracting it from 34-9 = 25 we get the number of primes that can be in this triple sequence. It's worth noting that the products of numbers greater than 3 and the number 3 grow like the squares of three consecutive numbers 3 ^ + 4 ^ = 5 ^, 9 + 16 = 25.
4 In the arithmetic sequence, the middle of the three consecutive terms is the arithmetic mean of both outer terms (89 + 101)/2 = 190/2 = 95. And the arithmetic mean of the difference of both outer terms shows the constant difference between the terms (101 - 89)/2 = 12/2 = 6, 89 -6- 95 -6-101. And this rule applies to the ratio of the number of products of numbers greater than 3 to the number of prime numbers (9 + 25)/2 = 34/2 = 17, (25 - 9)/2 = 16/2 = 8, 9 -8- 17 -8- 25. This tells us that the products of numbers greater than 3 are on average eight less (17 - 8 = 9), and prime numbers 8 more than the arithmetic mean of their sum (17 + 8 = 25), which is (25 + 9)/2 = 17. All this results from the properties of the parity of numbers, which says: the sum and difference of two numbers with the same parity is an even number; because 2k ± 2Ꙇ = 2 (k ± Ꙇ), (2k + 1) + (2Ꙇ + 1) = 2 (k + Ꙇ + 1) and (2k + 1) - (2Ꙇ + 1) = 2 (k - Ꙇ ), and the sum and difference of two numbers with different parity is an odd number; because 2k + (2Ꙇ + 1) = 2 (k + Ꙇ) + 1 and 2k - (2Ꙇ + 1) = 2 (k - Ꙇ) - 1 and (2k + 1) ± 2Ꙇ = 2 (k ± Ꙇ) + 1. It is true that the sum and difference of two numbers with the same parity are even numbers, but the arithmetic means of the sum and difference of two numbers with the same parity are numbers with different parity. And we will use this property to calculate the greater and lesser components of the sum and the difference. (a + b)/2 + (a - b)/2 = a, when a> b, (25 + 9)/2 + (25 - 9)/2 = 34/2 + 16/2 = 17 + 8 = 25, or (a + b)/2 - (a - b)/2 = b, when b <a, (25 + 9)/2 - (25 - 9)/2 = 34/2 - 16/2 = 17 - 8 = 9. We clearly see here that the number of primes and their products is not accidental from the very beginning, but follows from the properties of the numbers and all the laws of arithmetic. And this is what it looks like for the specific odd numbers we find up to 100.
5 There will be no more to equate the number of primes with their products of 25/25 = 1: 1. Arranged among their products in a strictly defined ratio 17/23, 17/33, 17/43, 17/53. Follow the tendency that their products are more and more π (340) = 68/102/170 = 2: 3: 5, also in other ratios π (1080) = 180/360/540 = 1: 2: 3, π (8440 ) = 1055/3165/4220 = 1: 3: 4. This equation of the ratio 34/16 to 25/25 is done by subtracting and adding the same number 34 = 9 + 25 - 9 = 16 to the arithmetic mean of the sum of the numbers (34 + 16) / 2 = 50/2 = 25, i.e. subtracting too much from 34 and adding it to 16/34 - 9 = 25 = 9 + 16 /.
PRIMARY NUMBER FUNCTION π(x) Until now, the prime numbers have seemed to be quite randomly distributed among other numbers. However, the number of prime numbers is constantly smaller the further areas we consider, and their number is decreasing both to half the number of numbers in a given quantity and their products π(100) = 25:25:50, 1/1/2, π(340) = 68: 102: 170, 2/3/5. As far as their distribution is concerned, prime numbers follow one rule that the sum of prime numbers and their products make up half of a given quantity /π(x) + ∑[p(p')] = ½N/, i.e. they are mutually dependent. Interestingly, the number of quantities of their products /∑p(p') / is always a number with the same parity as the number of prime numbers π(x). (25 + 25 = 50, 168 + 332 = 500, 1229 + 3771 = 5000, 9592 + 40408 = 50000, 78498 + 421502 = 500000, ....) The sum and difference of two numbers with the same parity is always even, and therefore divisible by two. The rule of half the sum and the difference of primes and their products allows us to calculate the number of primes up to half of a given quantity, because this half consists of half the sum and the difference of numbers with the same parity. Theorem: If the number of quantities of their products /∑p(p')/, is the same parity as the number of quantities of primes π(x), then half of their sum and difference added when their products are more than prime numbers, or subtracted when there are fewer primes than their products, it gives the exact value of π(x) to half of the given quantity.
6 Proof: Up to 1000 we have 168 primes and 332 primes. Half of the sum and difference of numbers with the same parity summed when their products are more than primes [332 + 168]/2 + [332 - 168]/2 = 500/2 + 164/2 = 250 + 82 = 332, or subtracted when there are fewer primes than their products [332 + 168]/2 - [332 - 168]/ 2 = 250 - 82 = 168, gives the exact value of π (x) to half of the given quantity. 168-number long sequence of primes from 2 - 997, completed by 166 products of 3 (9 - 999), and 166 products of numbers greater than 3 [168 + (166 + 166)] = 168 + 332 = 500, halfway given quantity. From this layout it is clear that the products of the number 3, which increases as half of a given quantity in a predictable geometric progression 16, [166 = 16(10) + 6, 1666 = 166(10) + 6], have an impact on the number of primes and their products greater than 3. That is, according to the basic ratio, there are 34 = 25 prime numbers plus 9 of their products greater than 3. /16 + 34 = 50 = 25 + (16 + 6 +3)/.
We can see that the mathematical laws resulting from the parity of numbers apply and in this equation (N)/2 - ∑[p(p')] = π(x), when we subtract from half of a given quantity the number of products of prime numbers it contains to obtain number of primes up to a given quantity: where the sum and difference of two numbers with the same parity is an even number / 500 - 332 = 168 /, and an odd number with different parity /50 - 25 = 25, 5000 - 3771 = 1229/.
7 N/2 10/2 10²/2 10³/2 10⁴/2 10⁵/2 10⁶/2 10⁷/2 10⁸/2 10⁹/2 10¹⁰/2 10¹¹/2 10¹²/2 10¹³/2 10¹⁴/2 10¹⁵/2 10¹⁶/2 10¹⁷/2 10¹⁸/2 10¹⁹/2 10²⁰/2 10²¹/2 10²²/2 10²³/2 10²⁴/2 10²⁵/2 10²⁶/2 10²⁷/2 10²⁸/2 10²⁹/2 10³⁰/2
-
∑[p(p‘)] 1 25 332 3 771 40 408 421 502 4 335 421 44 238 545 449 152 466 4 544 947 489 45 881 945 187 462 392 087 982 4 653 934 463 161 46 795 058 249 198 470 155 429 577 331 4 720 761 658 966 075 47 376 442 842 345 767 475 260 045 712 259 140 4 765 942 332 723 655 393 47 779 180 397 439 081 160 478 872 730 513 981 268 072 4 798 532 713 310 684 093 710 48 074 679 608 393 196 031 077 481 564 400 232 650 799 132 134 4 823 153 690 600 856 230 588 320 48 300 753 249 127 562 858 672 397 483 647 539 573 158 319 553 572 601 4,842,410,858,973,345,227,554,411,713 48,479,305,343,444,685,404,228,449,480 485,307,663,117,409,461,265,505,043,498
= π(x) 4 25 168 1 229 9 592 78 498 664 579 5 761 455 50 847 534 455 052 511 4 118 054 813 37 607 912 018 346 065 536 839 3 204 941 750 802 29 844 570 422 669 279 238 341 033 925 2 623 557 157 654 233 24 739 954 287 740 860 234 057 667 276 344 607 2 220 819 602 560 918 840 21 127 269 486 018 731 928 201 467 286 689 315 906 290 1 925 320 391 606 803 968 923 18 435 599 767 349 200 867 866 176 846 309 399 143 769 411 680 1 699 246 750 872 437 141 327 603 16 352 460 426 841 680 446 427 399 157 589 141 026 654 772 445 588 287 1,520,694,656,555,314,595,771,550,520 14,692,336,882,590,538,734,494,956,502
This leads to a balanced distribution of prime numbers and their products greater than 3 in the geometric sequence 3 (q) in the numbers 4 - 30 - 34 - 300 - 334 - 3000 - 3334. (34 = 9 + 25, 334 = 166 + 168, 3 334 = 2 105 + 1 229), that is [∑[p(p')]> 3 + π(x)] always completed to ½ N by the products of the number 3(16, 166, 1 666), as a sum of two constant numbers 34 + 16 = 50, 334 + 166 = 500, 3 334 + 1 666 = 5,000, and this sum grows in the geometric sequence of 5(q). This makes the half of the sum of the prime numbers and their products greater than 3, (25 + 9)/2 = 34/2 = 17 one number subtracted or added from half of the difference between them / 25 - 9 = 16/2 = 8 / , you can equate these two numbers 9 + 8 = 17 = 25 - 8, 168 - 166 = 2/2 = 1, 166 + 1 = 167 = 168 - 1, 2105 - 1229 = 876/2 = 438, 2105 - 438 = 1667 = 1229 + 438 half the sum of primes and their products greater than 3. So while there are less than half the sum of primes and their products greater than 3, there are more primes greater than 3. Hence, we can write this system further: ∑[p(p')> 3 = [π(x) + ∑p(p')> 3]/2 ± [π(x) - ∑p(p')> 3]/2 = π(x), 9 = (25 + 9) / 2 ± (25 - 9) / 2, 9 = 17 ± 8 = 25, 166 = 167 ± 1 = 168, 2105 = 1667 ± 438 = 1 229
8
So there is a close relationship between the number of arriving primes π(x) and their products greater than three /∑[p(p')]> 3 /, which increase exponentially 3(q), as can be seen in the table below. /3(q) = ∑[p(p')]> 3 + π(x), 30 = 9 + 21, 300 = 157 + 143,/ That is, if there are 4 prime numbers in the top ten, then up to 100 there cannot be more than 3(10) = 30, i.e. 21 primes plus 9 of their products greater than 3, this equals 30. 4 + 30 = 34 + 300 = 334 + 3000 = 3334, ∑p + [p(p’)]> 3 4 30
∑[p(p’)]> 3 0 9
π(x) 4 21
N 10
34
9 157 166
25 143 168
10²
300 334
10³
9 3 000
300 000 000 000 000 000 000
1 939 2 105 21 637 23 742 231 094 254 836 2 413 919 2 668 755 24 903 124 27 571 879 254 913 921 282 485 800 2 595 795 023 2 878 280 823 26 336 997 698 29 215 278 521 266 510 142 795 295 725 421 316 2 691 542 375 179 2 987 267 796 495 27 141 123 786 037 30 128 391 582 532 273 360 371 328 133 303 488 762 910 665 2,750,606,229,388,744 3 054 094 992 299 409 27,655,681,183,379,692 30,709,776,175,679,101 277,883,602,869,913,373 308,593,379,045,592,474 2,790,682,287,011,396,253 3,099,275,666,056,988,727 28,013,238,064,715,425,767 31,112,513,730,772,414,494 281,093,550,116,542,186,912
1 061 1 229 8 363 9 592 68 906 78498 586 081 664 579 5 096 876 5 761 455 45 086 079 50 847 534 404 204 977 455 052 511 3 663 002 302 4 118 054 813 33 489 857 205 37 607 912 018 308 457 624 821 346 065 536 839 2 858 876 213 963 3 204 941 750 802 26 639 628 671 867 29 844 570 422 669 249,393,770,611,256 279 238 341 033 925 2,344,318,816,620,308 2 623 557 157 654 233 22,116,397,130,086,627 24 739 954 287 740 860 209,317,712,988,603,747 234 057 667 276 344 607 1,986,761,935,284,574,233 2 220 819 602 560 918 840 18,906,449,883,457,813,088
333 333 333 333 333 333 334
312,206,063,847,314,601,406
21 127 269 486 018 731 928
3 000 000 000 000 000 000 000
2,819,659,982,796,702,825,638
180,340,017,203,297,174,362
3 333 333 333 333 333 333 334
3,131,866,046,644,017,427,044
201 467 286 689 315 906 290
30 000 000 000 000 000 000 000
28,276,146,895,082,511,937,367
1,723,853,104,917,488,062,633
33 333 333 333 333 333 333 334
31,408,012,941,726,529,364,411
1 925 320 391 606 803 968 923
300 000 000 000 000 000 000 000
283,489,720,624,257,603,101,057
16,510,279,375,742,396,898,943
3 334 30 000 33 334 300 000 333 334 3 000 000 3 333 334 30 000 000 33 333 334 300 000 000 333 333 334 3 000 000 000 3 333 333 334 30 000 000 000 33 333 333 334 300 000 000 000 333 333 333 334 3 000 000 000 000 3 333 333 333 334 30 000 000 000 000 33 333 333 333 334 300 000 000 000 000 333 333 333 333 334 3 000 000 000 000 000 3 333 333 333 333 334 30 000 000 000 000 000 33 333 333 333 333 334 300 000 000 000 000 000 333 333 333 333 333 334 3 000 000 000 000 000 000 3 333 333 333 333 333 334 30 000 000 000 000 000 000 33 333 333 333 333 333 334
333 333 333 333 333 333 333 334
314,897,733,565,984,132,465,468
18 435 599 767 349 200 867 866
3 000 000 000 000 000 000 000 000
2,841,589,290,368,205,431,456,186
158,410,709,631,794,568,543,814
3 333 333 333 333 333 333 333 334
3,156,487,023,934,189,563,921,654
176 846 309 399 143 769 411 680
30 000 000 000 000 000 000 000 000
28,477,599,558,526,706,628,084,077
1,522,400,441,473,293,371,915,923
33 333 333 333 333 333 333 333 334
31,634,086,582,460,896,192,005,731
1 699 246 750 872 437 141 327 603
300 000 000 000 000 000 000 000 000
285,346,786,324,030,756,694,900,204
14,653,213,675,969,243,305,099,796
333 333 333 333 333 333 333 333 334
316,980,872,906,491,652,886,905,935
16 352 460 426 841 680 446 427 399
3 000 000 000 000 000 000 000 000 000
2,858,763,319,400,186,848,000,839,112
141,236,680,599,813,151,999,160,888
3 333 333 333 333 333 333 333 333 334
3 175 744 192 306 678 560 887 745 047
157 589 141 026 654 772 445 588 287
10⁴ 10⁵ 10⁶ 10⁷ 10⁸ 10⁹ 10¹⁰ 10¹¹ 10¹² 10¹³ 10¹⁴ 10¹⁵ 10¹⁶ 10¹⁷ 10¹⁸ 10¹⁹ 10²⁰ 10²¹ 10²² 10²³ 10²⁴ 10²⁵ 10²⁶ 10²⁷ 10²⁸
10
RATIO [π(x) + ∑[p(p’)> 3]/π(x)
N
[π(x) + ∑[p(p’)]>3]/π(x)
Q
Q₂ - Q₁ = d
10 10² 10³ 10⁴ 10⁵ 10⁶ 10⁷ 10⁸ 10⁹ 10¹⁰ 10¹¹ 10¹² 10¹³ 10¹⁴ 10¹⁵ 10¹⁶ 10¹⁷ 10¹⁸ 10¹⁹
4 34/25 334/168 3 334/1 229 33 334/9 592 333 334/78 498 3 333 334/664 579 33 333 334/5 761 455 333 333 334/50 847 534 3 333 333 334/455 052 511 33 333 333 334/4,118,054,813 333 333 333 334/37,607,912,018 3 333 333 333 334/346,065,536,839 33 333 333 333 334/3,204,941,750,802 333 333 333 333 334/29,844,570,422,669 3 333 333 333 333 334/279,238,341,033,925 33 333 333 333 333 334/2,623,557,157,654,233 333 333 333 333 333 334/24,739,954,287,740,860
1,36 1,988095 2,712775 3,475118 4,246464 5,015707 5,785575 6.555545 7.325161 8.094436 8.863383 9.632086 10.400605 11.168977 11.937233 12.705396 13.473482
3 333 333 333 333 333 334/234,057,667,276,344,607
14.241504
0,628095 0,724679 0,762412 0,771283 0,769243 0,769970 0,769875 0.769616 0.769275 0.768947 0.768703 0.768519 0.768372 0.768256 0.768163 0.768086 0.768022
10²⁰ 10²¹ 10²² 10²³ 10²⁴ 10²⁵ 10²⁶ 10²⁷ 10²⁸ 10²⁹ 10³⁰
33 333 333 333 333 333 334/2,220,819,602,560,918,840
15.009473
0.7679691
333 333 333 333 333 333 334/21127269,486,018,731,928
15,777397
0,7679234
3333 333 333 333 333 333 334/201467286689315906290
16,545283
0,767886
33 333 333 333 333 333 333 334/1925320391606803968923
17,313135
0,767852
333333333333333333333334/18435599767349200867866
18,0809595315
0,767824
3333333333333333333333334/176846309399143769411680
18,848758250
0,767799
33333333333333333333333334/1699246750872437141327603
19,616534983
0,767776733
333333333333333333333333334/16352460426841680446427399
20,3842923102987409
0,7677573269
333333333333333333333333333334/157589141026654772445588287
21,1520496375415249
0,767757327242784
33333333333333333333333333334/1520694656555314595771550520
21.9198069708748582
0,767757333333333
333333333333333333333333333334/14692336882590538734494956502
22,68756400000000
0,767757029125141
The constant increase in the sum of prime numbers and their products greater than three by 3 (q) causes that also the ratio of π (x) to the sum of primes and their products greater than 3 constantly increases by almost the same value 0.76. From this uniformly increasing sequence of primes and their products greater than 3, exponentially 3 (q), giving a total of three and one four, with the ratio of prime numbers to this sum constantly increasing by 0.7677573. When it’s 20.3842923102987409169159173364374662155816378 + 0.7677573 = 21.1520496375415249169159173364374662155816378 and by this ratio, the known sum of prime numbers and their products greater than 3 divided by us gives the number of primes up to 10 ²⁸.
11
By subtracting from the sum [π(x) + ∑[p + p(p')]> 3], the number of primes we get the number of their products greater than 3. It is a strict relationship of the number of prime numbers with their products greater than 3, always to the number growing exponentially 3(q) /4 - 30 - 34 - 300 - 334 3000 - 3334/, testifies to a wonderful order in the whole sequence of primes from the very beginning, like in the best accounting book, where everything must match zero as seen below. π(x) + d = π(x'), 4 + 21 = 25, 25 + 143 = 168, 168 + 1061 = 1229, 1229 + 8363 = 9592, 9592 + 68906 = 78498, 78498 + 586081 = 664579, 664579 + 5,096,876 =
5 761 455, 5 761 455 + 45 086 079 = 50 847 534, 50 847 534 + 404 204 977 = 455 052 511, 455 052 511 + 3 663 002 302 = 4 118 054 813, 4 118 054 813 + 33 489 857 205 = 37 607 912 018, 37 607 912 018 + 308 457 624 821 = 346 065 536 839, 346 065 536 839 + 2 858 876 213 963 = 3 204 941 750 802,
12
3 204 941 750 802 + 26 639 628 671 867 = 29 844 570 422 669, 29 844 570 422 669 + 249 393 770 611 256 = 279 238 341 033 925, 279 238 341 033 925 + 2 344 318 816 620 308 = 2 623 557 157 654 233, 2 623 557 157 654 233 + 22 116 397 130 086 627 = 24 739 954 287 740 860, 24 739 954 287 740 860 + 209 317 712 988 603 747 = 234 057 667 276 344 607,
13
21 127 269 486 018 731 928 + 180 340 017 203 297 174 362 = 201 467 286 689 315 906 290,
201 467 286 689 315 906 290 + 1 723 853 104 917 488 062 633 = 1925320391606803968923 + 16,510,279,375,742,396,898,943 = 18,435,599,767,349,200,867,866 + 158,410,709,631,794,568,543,814 = 176,846,309,399,143,769,411,680 + 1,522400,441,473,293,371,915,923 = 1,699,246,750,872,437,141,327,603 + 14,653,213,675,969,243,305,099,796 = 16,352,460,426,841,680,446,427,399 + 141,236,680,599,813,151,999,160,888 = 157,589,141,026,654,772,445,588,287.
14
And this is what it looks like, collected in one table, showing exactly how to calculate the specific components of the sum [π(x) + ∑[p(p’)> 3].
15
Everything must be in agreement here. [π(x) + ∑p(p')> 3]/2 ± [π(x) - ∑p(p')> 3]/2 = π(x) = ∑p (p')>3, only π(10²⁹) equal 1,520,694,656,555,314,595,771,550,520 = 16,666,666,666,666,666,666,666,666,666,667 ± 15,145,972,010,111,352,070,895,116,147 = 31,812,638,676,778,018,737,561,782,814 and π(10³⁰) equal 14,692,336,882,590,538,734,494,956,502 = 166,666,666,666,666,666,666,666,666,667 ± 151,974,329,784,076,127,932,171,710,165 = 318,640,996,450,742,794,598,838,376,832 equation that makes sense.
16
The sum of these numbers grows exponentially 3 (q), hence the ratio of prime numbers to their sum with products greater than 3 is constantly increasing by 0.767757.
Thus, the puzzle of the distribution of prime numbers has been solved. Henceforth the sequence of primes is not like a random sequence of numbers, but rather a sequence of 3(q) increasing order of prime numbers and their products greater than 3, together with the products of 3 to half a given quantity. So the sum of the prime numbers and their products greater than three π(x) + ∑p(p')> 3 equals the difference between a half of a given quantity and the products of three ½N - ∑p(3), 25 + 9 = 34 = 50 - 16 and grows exponentially 3(q), 34 - 300 - 334 - 3000 - 3334 In total, on the radar chart this gives a picture of a grid of numbers arranged radially in 29 columns with a constant spacing n (58) between them, although you can also see 2 lefthanded (products of 7) and 8 right-handed (primes and 2 products of 5) spiral structures , as well as 4 smaller clockwise spirals of the products of 17 (289-493-697-901). Finally, the mysterious structure of prime numbers and their products, sought for centuries by mathematicians, has been discovered and its music can be written endlessly.
Q
E
D
17
TABLES π(x) and ∑[p(p’)>3] from 10 to 10³⁰
18
19
TABLES OF FIRST NUMBERS FROM 2 – 13 577
20
21
22
23
24
25
26
27
