Prime numbers - myths and facts

lubinaj7@gmail.com

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ABSTRACT There is a common belief bordering on the certainty that prime numbers are arranged on a chaotic number line, and that they are not governed by any laws that would allow us to describe them all clearly. In this article, "Prime numbers - myths and facts", I will try to show what laws govern and what order prevails among prime numbers, which allows me to clearly describe them. Whoever reads the article to the end will find out what unusual order and harmony prevails in the world of primes instead of chaos. “There are secrets that the human mind can never penetrate. To find out, just take a look at the prime number tables and you should see that there are neither order nor rules.

"Mathematicians have tried in vain to find some order in the sequence of prime numbers, and we have reason to believe that this is a mystery that the human mind will never fathom."

Leonard Euler – 1751

MYTHS Prime numbers are the very atoms of arithmetic. They also embody one of the most tantalising enigmas in the pursuit of human knowledge. How can one predict when the next prime number will occur? Is there a formula which could generate primes? These apparently simple questions have confounded mathematicians ever since the Ancient Greeks.

In 1859, the brilliant German mathematician Bernard Riemann put forward an idea which finally seemed to reveal a magical harmony at work in the numerical landscape. The promise that these eternal, unchanging numbers would finally reveal their secret thrilled mathematicians around the world. Yet Riemann, a hypochondriac and a troubled perfectionist, never publicly provided a proof for his hypothesis and his housekeeper burnt all his personal papers on his death.

Whoever cracks Riemann's hypothesis will go down in history, for it has implications far beyond mathematics. In business, it is the lynchpin for security and e-commerce. In science, it has critical ramifications in Quantum Mechanics, Chaos Theory, and the future of computing. Pioneers in each of these fields are racing to crack the code and a prize of $1 million has been offered to the winner. As yet, it remains unsolved.

So much for the myths about primes contained in the breathtaking book Music of Prime Numbers, where mathematician Marcus du Sautoy tells the story of eccentric and brilliant people who struggled to unravel one of the greatest mysteries of science.

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PREFACE "It will be millions of years before we’ll have any understanding, and even then it won’t be a complete understanding, because we’re up against the infinite."

P. Erdös (interview with P. Hoffman - Atlantic Monthly, November 1987, p. 74) "The sequence of primes has an imperceptible pattern, and as such, primes are a law to themselves. Although they appear to be like wild weeds scattered among natural numbers, ... For centuries mathematicians have tried and failed to explain what the basic formula of primes is. It is possible that there is no such pattern and the prime numbers by their nature show a random arrangement, in which case mathematicians are advised to undertake other less ambitious questions in this field. " Simon Singh This was the case before the discovery of the regular formula π(N) + Σ[p (p')] = ½N saying that half of a given quantity is the sum of primes and their products in a strictly defined ratio, hidden behind seemingly chaotically arranged primes, and in my work I tried to add an explanation to this. I have shown that p = [a + (a + 1)]/1 is the only pattern that is inseparable from prime numbers, because they are not randomly distributed, but due to the congruence modulo 7 they have an even number as its predecessor, half of which ensures that all preceding numbers form pairs of extreme terms with identical intermediate sums, having no common divisor greater than 1, which proves that they are prime numbers. All of this gives us the proverbial net to capture the remaining unresolved issues, such as the ratio of primes to products and half a given quantity (½N), the ratio of twin primes to prime, and the strong and weak guess of Goldbach, structure of primes and many others.

Prime numbers are the subject of more attention to mathematicians, both professional and amateur, since people began to study the properties of numbers and find them fascinating.

For example, Euclid has already shown that there are infinitely many primes. However, several important properties of prime numbers are not yet well understood. For centuries, prime numbers have troubled curious thinkers. On the one hand, prime numbers appear to be randomly distributed among natural numbers with no law other than probability.

However, on the other hand, the distribution of global primes reveals remarkably smooth regularity. This combination of randomness and regularity motivated me to look for patterns in the arrangement of prime numbers that might eventually shed light on their ultimate nature. By writing this work, I wanted to synthesize what is already known about prime number theory and show it as a field in which the natural problems of number theory are systematically examined as a whole. I hope all math lovers will feel happy reading these pages.

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"If you can't rethink everything, you'll never change anything that already exists."

Anwar al Sadat FACTS We have known this, seemingly chaotic prime number, for over 2,000 years. However, when looking at the line graph, we see that the prime numbers follow in the correct order. From the top ten, prime numbers take the four characteristic numbers of units 11, 13, 17, 19, among which the constant distance is 11 - 6 - 17, 13 - 6 - 19.

Let us briefly look at the following line diagram of the sequence of primes and their products up to 100. At first glance, we notice a repeating clear diagonal sequence in green, every three numbers of the products of 3, with a constant interval of 6 (9 - 15 - 21 - 27 - 33 ,). The same pink and blue sequence of prime numbers and their products greater than 3 (2 - 7 - 13 - 19 – 25/5) and (5 - 11 - 17 - 23 - 29 – 35/7) are interwoven between them.

Together they form an evenly expanding arithmetic sequence starting from the number 3, each term of which (starting with the second) is formed by adding the previous constant number (r = 2) to the term. Such a sequence as a set of all odd numbers is an infinite sequence. We will find out in a moment what influence this has on the arrangement of prime numbers and their products.

5 BASIC ORDER So let's break this arithmetic sequence of prime numbers and their products into three basic sequences, as in the table below.

And what do we see? Here are the next four primes (2,3,5,7) complemented by the product of the number three (9) form the first system according to the formula π(N) + ∑p(p') = ½N, which says that the prime numbers are complemented by their products up to half the given amount. 4 + 1 = 5 to ten we have 4 primes and one product of three or 9, which makes them half of the given quantity 10/2 = 5 in the given proportion (4: 1). The prime numbers are completed in the same proportion to 20. We have here eight primes (2,3,5,7,11,13,17,19) and two products of the number 3 (9,15), i.e. (8 + 2)/2 = 10/2 = (4: 1).

6 However, to 96, the proportions of prime numbers and their products are equal to (1: 1). To 96 we have 24 primes (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79 , 83, 89), 15 products of 3 (9 - 93) and 9 products of numbers greater than 3 (25/5, 35/7, 55/11, 65/13, 85/17, 95/19, 49/7 , 77/11, 91/13) or 24 + (15 + 9) = (24 + 24)/24 = 48/24, (1: 1). The same applies to 100, where for 25 primes there are 16 products of 3 (9 - 99) plus 9 products of numbers greater than 3, i.e. 25 + (16 + 9) = (25 + 25)/25 = 50/25, (1: 1).

We know that any integer greater than one, divisible only by 1, and itself is prime. This definition implies their property that they are decomposed into the sum p = [a + (a + 1)]/1 of two relatively prime numbers whose greatest common divisor is the number one. In this approach, they are the halves of the preceding and one greater half of the following even number, which are always relatively prime numbers, as can be seen in the table below.

2/2, 3, 4/2, (1 + 2) / 1 = 3/1, 4/2, 5, 6/2, (2 + 3) / 1 = 5/1, 6/2, 7, 8/2, (3 + 4) / 1 = 7/1, 10/2, 11, 12/2, (5 + 6) / 2 = 11/1, 12/2, 13, 14/2, (6 + 7) / 1 = 13/1, 16/2, 17, 18/2, (8 + 9) / 1 = 17/1, 18/2, 19, 20/2, (9 + 10) / 1 = 19/1, 22/2, 23, 24/2, (11 + 12) / 1 = 23/1, 28/2, 29, 30/2, (14 + 15) / 1 = 29/1. Also the other components of intermediate sums are relatively prime numbers: (1 + 4) / 1 = 5, (2 + 5) / 1 = 7/1, (1 + 6) / 1 = 7/1, (1 + 10) / 1 = 11/1, (2 + 9) / 1 = 11/1, (3 + 8) / 1 = 11/1, (4 + 7) / 1 = 11/1, (5 + 6) = 11.

Hence, we can write a general formula for each prime number that is the sum of the extreme pairs of preceding relatively prime numbers that make up identical intermediate sums: [(n – 1)/2 + (n + 1)/2]/1 = p/1

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8 But not all odd numbers are prime numbers, and although they are formed on the same principle, when decomposing them we will see that they not only consist of the extreme pairs of relatively prime numbers preceding them, but also numbers whose highest common divisor is a number greater than one. Then we deal with complex numbers (9/3, 15/5, 21/7, 25/5, 27/3, 33/11, 35/7, 39/13, 45/5, 49/7, 51/17), because 9 = (4 + 5)/1, but also 9 = (3 + 6)/3, 51 = (25 + 26)/1, 51 = (17 + 34)/17.

According to the additive number theory, each odd number can be represented as the sum of two different components of the extreme preceding numbers, forming identical intermediate sums, having no common divisor greater than one and then it is prime, e.g .: 7 = (6 + 1)/1 = (5 + 2)/1 = (4 +3) /1, or having a common divisor greater than 1 and then complex, e.g. 9 = (8 + 1)/1 = (7 + 2)/1 = (6 +3) /3 = (5 + 4)/1. We can see that there are always half of the preceding even number such distributions that make up identical intermediate sums, so 6/2 = 3 to 7, 8/2 = 4 to 9, 10/2 = 5 to 10. In this way we can easily calculate the sum of all the preceding numbers by multiplying the identical sums equal to the given number by the half of the preceding even number, e.g .: 3 (7) = 21, 4(9) = 36, 5(11) = 55.

Decomposing these numbers into prime factors is irrefutable proof that a given number consists of all prime numbers 21/3 = 7, 3(7) = 21, 55/5 = 11, 5(11) = 55, or complex 36/3 = 12/3 = 4/2 = 2, 3(3) = 9, 2(2) = 4, 4(9) = 36. Realizing that adding pairs of words from opposite ends of a list of numbers preceding an odd number always yields identical subtotals (6 + 5) = 11 = (7 + 4), tells us whether the given triangular number, as the sum of the numbers preceding the given quantity, consists only of prime numbers (55/5 = 11) or complex numbers (36/4 = 9).

If the sum of the preceding numbers, i.e. a given triangular number, primes up to a given number, it means that each pair of components does not have a common divisor greater than one and the given number is prime. Factoring a given triangular number into prime factors less than a given number means that at least one pair of terms has a common divisor greater than one and the given number is

9 composite. 9 = (8 + 1)/1 = (7 + 2)/1 = (6 + 3)/3 = (5 + 4)/1, 4(9) = 36/2 = 18/2 = 9/3 = 3/3 = 1, (2*2)(3* 3) = 36, so the number 9 is a composite number. The number 11 is prime because the five pairs of identical sums that make it up, added outwardly as the numbers preceding the given number, do not have a common divisor greater than 1 and in total give 55, a triangular number completely divisible by the number of identical intermediate sums, equal to half the number in front of it. an even number. (10 + 1)/1, = (9 + 2)/1, = (8 + 3)/1, = (7 + 4)/1, = (6 + 5)/1. 5(11) = 55/5 = 11

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Triangular numbers 3, 10, 21, 36, 55, ... as a sum of numbers preceding a given odd number, consist of n - the number of pairs of terms of added words from opposite ends of the list of preceding numbers, equal to half of the preceding even number 2/2, 4/2, 6/2, 8/2, 10/2, which, if they do not have a common divisor greater than 1, form identical intermediate sums of only prime numbers (4 + 1)/1, (2 + 3)/1, 5 + 5 = 10/2 = 5, and if they have at least one common divisor greater than 1, they form identical intermediate sums only of complex numbers (8 + 1)/1, (7 + 2)/1, (6 + 3)/3, (5 + 4)/1.9 + 9 + 9 + 9 = 36/4 = 4 * 9 = (2 * 2) (3 * 3).

This systematic process of determining which number is the product of prime or prime numbers as seen in the table above is a good example of this testing algorithm: [(n - 1) (n)] / 2 = n / 2 = t | p = (n = p) or (n) / 2 = t | p = (p <n) = p (p ').

It is based on the fundamental property of prime numbers to form n - the number of pairs of components with identical intermediate sums that do not have a common divisor greater than 1. Then the triangular number as the sum of all the preceding numbers decomposes into prime factors up to the given number, which means that is prime. When it decomposes prime factors less than a given number, it is a composite number. An algorithm is a method by which we can solve a problem by following its instructions. When we apply this, then we have an irrefutable certificate confirming that the given number is a prime number or their product. [(n-1)(n)]/2 [36(37)]/2 [44(45)]/2 [94(95)]/2 [100(101)]/2

n/2 1332/2 1980/2 8930/2 10100/2

t 666 990 4465 5050

t/p 666/2 990/11 4465/47 5050/5

n/p 333/3 90/3 95/19 1010/5

n/p 111/3 30/5 5/1 202/2

n = p, (p<n) 37 = p 6/3 = 2 (p<n) 5 (p<n) 101 = p

[(n â&#x20AC;&#x201C; 1)/2](n) = t | p = (n = p), or t | p = (p < n) = p(pâ&#x20AC;&#x2122;)

Let's check the properties of the number (1,378,565,437 - 1)/2, from which we subtract 1 and divide by 2, to see how many pairs of components with identical intermediate sums are formed, which is equal to 1,378,565,436/2 = 689,282,718 pairs and multiply by this number to get the sum of all preceding numbers, i.e. a triangular number t = 950,221,331,356,217,766, which decomposes into prime factors up to a given number, i.e. a given number is prime, because it consists of 689,282,718 pairs of components of identical intermediate sums of a prime number 1,378,565,437.

1,378,565,437(689,282,718) = 950,221,331,356,217,76

(1 + 1,378,565,436)/1 = 1,378,565,437

950,221,331,356,217,766/3 = 316,740,443,785,405,922

(2 + 1,378,565,435)/1 = 1,378,565,437

316,740,443,785,405,922/2 = 158,370,221,892,702,961

(3 + 1,378,565,434)/1 = 1,378,565,437

158,370,221,892,702,961/114,880,453 = 1,378,565,437

(4 + 1,378,565,433)/1 = 1,378,565,437

114,880,453*6 = 689,282,718

(689,282,718 + 689,282,719)/1 = 1,378,565,437

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"Prime numbers are famous for creating an impenetrable tangle. According to many mathematicians, their order does not follow a discernible pattern. " Vine Guy HOW ARRANGEMENTS ARE THE PRIME NUMBERS

A better understanding of prime numbers, for a mathematician, hopes to find new paths through the overwhelming complexity of the world of mathematics. Despite their apparent simplicity and essential character, prime numbers were the most mysterious objects mathematicians have studied. Questions about the distribution of prime numbers were among the most difficult. These were questions of a purely theoretical nature for a long time, but today the prime numbers have found application in various fields.

Suddenly, there is also an economic interest in the question of whether the evidence of Riemann's conjecture can tell us anything about the distribution of prime numbers in the world of numbers. For centuries, a magic formula has been searched in vain, it may be time to come up with a list of primes, so it's time to approach the matter with a new strategy. So far the prime numbers have seemed to appear quite by chance. Of course, such an attitude does not allow us to predict what the prime number after 10,000 will be.

In fact, the distribution of prime numbers depends on the strict relation to its products, and this results from the ability to complete the primes by their products up to half of a given value, according to the formula π(N) + ∑p(p') = ½N, which says that the sum of the number of primes and their products equals half of the given quantity.

From 20 to 100 there are 80 numbers, half of which is 40 = 17 + 23 are prime numbers and their products. To 20/2 = 10 = 8 + 2 we have 8 prime numbers (2, 3, 5, 7, 11, 13, 17, 19) and 2 products (9, 15), and to 100/2 = 50 = [( 8 + 17) + (2 + 23)] = 25 + 25 the ratio becomes even.

The formula π(N) + ∑p(p')> 3 = 3(q) says that the number of primes up to a given quantity N plus the number of products of primes greater than 3 creates a constant sum, growing exponentially 3(q) . Also the products of the number 3 form a constant sum up to 100 is 16 of them. These two sums (34) + (16) = 50 complete half of the given quantity, which grows exponentially 5(q). Hence we write [π(N) + ∑p(p')> 3] + ∑3(p) = ½N, (25 + 9) + 16 = (34) + (16) = 50 and this is the basic distribution formula prime numbers. We can see that the ratio of primes to half a given quantity 50/25 is equal to 2: 1.

This proves the perfect order of the whole sequence of natural numbers, which consists of 50% even and odd numbers, i.e. prime numbers and their products. Such basic numbers are not determined by nature by the method of accidental tossing a coin or the "God does not play dice with the world", but based on the ability to create identical intermediate sums of

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the n - th number of pairs of extreme number components preceding a given quantity.

Chance and chaos are simply unacceptable to mathematics.

So it is enough to create a table from the numbers that follow regularly every 2 (3, 5, 7, 9, 11, 13, 15, 17, 19), and without Eratosthenes' sieve we have selected all primes and their products. At first glance you can see with what regularity there are products of number 5, every 30 numbers (25-55-85, 35-65-95) or products of number 7, every 28 numbers (49-77, 133-161), or products of 11, every 88 numbers (121-209, 187-275), or the products of 13, every 26 numbers (221-247, 377-403), or the products of 17, every 34 numbers (289-323).

This table clearly shows that prime numbers together with products greater than 3 form two separate sequences (11-41-71-101-131-161, 13-43-73-103-133), and in these sequences the prime numbers are supplemented by its products to half of the given quantity with exemplary accuracy [10(p) + 5p(p')]/5 = 15/5, 2 : 1, [16(p) + 12p(p')]/4 = 28/4, 4 : 3, 30(p) + 30p(p') = 120/2 = 60, 1: 1, ..

N/2 10/2 10²/2 10³/2 10⁴/2 10⁵/2 10⁶/2 10⁷/2

∑[p(p‘)] 1 25 332 3 771 40 408 421 502 4 335 421

π(N) 4 25 168 1 229 9 592 78 498 664 579

13 10⁸/2 10⁹/2 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

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

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

So there is a close relationship between the number of arriving primes π (N) and their products greater than three /∑ [p(p')]> 3/, which increase exponentially 3(q), as shown in the table below. /3(q) = ∑[p(p')]> 3 + π(N), 30 = 9 + 21, 300 = 157 + 143, ../ That is, if there are 4 prime numbers in the top ten, then to 100 there cannot be more than 3(10) = 30, i.e. 21 primes plus 9 of their products greater than 3, which is equal to 30, 4 + 30 = 34 + 300 = 334 + 3000 = 3334,.. ∑d(p) +∑d [p(p’)]> 3 = 3(q) 4

∑[p(p’)]> 3 0

π(x) 4

30

9 9 157 166 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

21 25 143 168 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

34 300 334 3 000 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

N 10 10² 10³ 10⁴ 10⁵ 10⁶ 10⁷ 10⁸ 10⁹ 10¹⁰

14 33 333 333 334

300 000 000 000 000 000 000

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

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

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

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

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

10¹¹ 10¹² 10¹³ 10¹⁴ 10¹⁵ 10¹⁶ 10¹⁷ 10¹⁸ 10¹⁹ 10²⁰ 10²¹ 10²² 10²³ 10²⁴ 10²⁵ 10²⁶ 10²⁷ 10²⁸

And the prime numbers and their products are evenly distributed, as they follow each other in 20 columns every 40-80 numbers and in rows with a constant number of places for primes and their products greater than 3, supplemented to ½N by the products of 3 (4 - 3 - 3) (6 - 7 7), i.e. 6 places in this line, 4 may be occupied by prime numbers, and 2 by their products greater than 3, similarly in lines by 7 places, 5 for prime numbers and 2 for their products greater than 3.

Only in the first row there is room for 8 primes, and since every third number in each row is the product of 3, there will be 2 places for the products of 3 in the next row, among 6 places, there are 4 places for the products of 3, and there are 7 places for 3 places, for the products of the number three.

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This leads to a balanced distribution of primes and their products greater than 3 included in the geometric sequence 3(q), 4 - 30 - 34 - 300 - 334 - 3000 - 3334. (34 = 9 + 25, 334 = 166 + 168, 3,334 = 2,105 + 1,229), i.e. ∑[p(p’)]> 3 + π(N) + ∑3(p) = ½N, always supplemented to ½N by the products of the number 3 (16, 166, 1666), as a sum of two constant numbers 34 + 16 = 50, 334 + 166 = 500, 3,334 + 1,666 = 5,000, and the sum grows exponentially 5(q).

As the table above shows, the number of prime numbers and their products greater than 3 increases steadily and steadily by 3(q), which results from the equation d[π(N)] + d[∑p(p')> 3] = 3(q) = [π(N') + ∑p(p')> 3] - [π(N) + ∑p(p')> 3], where the sum of the differences between the preceding number of prime numbers and their products greater than 3 is the difference of the sums: 25 - 4 = 21, 9 - 0 = 9, 21 + 9 = 30 = [25 + 9] - [4 +0] = 34 - 4, 168 - 25 = 143, 166 - 9 = 157, 143 + 157 = 300, 168 + 166 = 334, 25 + 9 = 34, 334 - 34 = 300, 1229 - 168 = 1061, 2105 166 = 1939, 1061 + 1939 = 3000, 1229 + 2105 = 3334, 3334 - 334 = 3000,

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This is what it looks like in an area chart, where the sum of the number of primes to a given quantity and their products greater than 3 is 50% of the total, and the ratio of products greater than 3 to primes increases asymptotically.

RATIO OF PRIMES TO PRODUCTS AND ½N Each natural number is either a prime number or a prime product, meaning that each integer uniquely decomposes into a product of prime numbers. Hence the mutual dependence of prime numbers on the number of integers, i.e. how many primes can be in a given range of numbers.

The ratio of prime numbers to products has already been determined, and it results from the ability to produce identical intermediate sums for a given quantity. Up to ten we have 4 primes (2 + 3 + 5 + 7 = 17), their sum is 17 and they form 4 identical intermediate sums up to 10 [(2 + 8 = 10), (3 + 7 = 10), (5 +5 = 10), (7 + 3 = 10), (8 + 7 + 5 + 3 = 23), and the sum of the items for 10 is 23. (17 + 23 = 40/4 = 10). According to this scheme, the ratio of prime numbers to their products will be shaped, i.e. 40 odd numbers in a given interval, there can be 17 primes (from 20 - 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 to 100) and 23 of their products (21, 25, 27, 33, 35, 39, 45, 49, 51, 55, 57, 63, 65, 69 , 75, 77, 81, 85, 87, 91, 93, 95, 99), which can be written by the formula π(N) + ∑[p(p')] = ½N, which says that the sum of the number of prime numbers and their products is equal to half of the given amount. From 20 to 100 there are 80 numbers, half of them (40 = 17 + 23) are 17 primes and 23 their products. To 20/2 = 10 = 8 + 2 we have 8 prime numbers (2, 3, 5, 7, 11, 13, 17, 19) and 2 products (9, 15), and to 100/2 = 50 = [( 8 + 17) + (2 + 23)] = 25 + 25 the ratio becomes even 1 : 1.

17

Up to 100/2 = 50 only 10 primes (3, 5, 7, 11, 13, 17, 19, 23, 29, 31) produce 25 products (9/3, 15/5, 21/7, 25/5 , 27/9, 33/11, 35/7, 39/13, 45/15, 49/7, 51/17, 55/11, 57/19, 63/7, 65/13, 69/23, 75 / 25, 77/11, 81/27, 85/17, 87/29, 91/13, 93/31, 95/19, 99/33). Each of these numbers knows its place in the series, which means that more than 25 products in these three sequences do not fit, because each product stands in a place which it signifies and is reserved only for him.

Such an even distribution of primes and their products greater than 3 makes it possible to find places where this ratio also grows proportionally with respect to ½N, as we can see in the area chart, and the ratio of products to primes is greater than the ratio of primes to half of a given quantity. ∑p(p’)/π(N)> π(N)/½N, 25/1 + 25/1 = 50/2, 1 : 1> 1 : 2

168/1 + 336/2 = 504/3, 1 : 2 > 1 : 3, 1055/1 + 3165/3 = 4220/4. 1 : 3 > 1 : 4, from this dependence and ordering, it is possible to calculate exactly what number of prime numbers is in a given interval, and also predict their number in the next interval of numbers up to half of a given quantity ½N.

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With the same accuracy as one, you can calculate the number of primes using their constant 1.151636 ratio of prime numbers to half of a given quantity ½N. This is because both prime numbers and their products form sequences, and the sum of the sequence is divisible by 16 π(N) + ∑p(p') = ½N/16, 9.592 + 40.408 = 50.000/16 = 3.125 when 9,592 primes are paired with their 40,408 primes, they form a sequence of 16(3,125) = 50,000 numbers. If we now divide the quotient of the sum of the prime numbers and their products by the quotient of the prime numbers themselves 9.592/16 = 599.5 we get 3.125/599.5 = 5.212677231, what is the ratio of prime numbers to half of the given quantity. Conversely, dividing the quotient of

19

all numbers by half of the given quantity by the ratio of prime numbers 3.125/5.212677231 = 599.5, we obtain the quotient of primes in this sequence, which multiplied by 16(599.5) = 9.592 gives the number of primes in the sequence with the accuracy one. The easiest way is 50,000/5.212677231 = 9.592, so ½N has been decoded.

31,250/6,369589 = 4,906.125(16) =78,498 p 312,500/7.52356 = 41,536.1875(16) = 664,579 p 3,125,000/8.6783633648 = 360,090.9375(16) = 5,761,455 p 31,250,000/9.8333185636 = 3,177,970.875(16) = 50,847,534 p 312,500,000/10.9877429 = 28,440,781.9375(16) = 455,052,511 p 3,125,000,000/12.1416548 = 257,378,425.8125(16) = 4,118,054,813 p 31,250,000,000/13.295 = 2,350,494,501.125(16) = 37,607,912,018 p 312,500,000,000/14.44813 = 21,629,096,052.4375(16) = 346,065,536,839 p 3,125,000,000,000/15.6009 = 200,308,859,425.125(16) = 3,204,941,750,802 p 31,250,000,000,000/16.75346 = 1,865,285,651,416.8125(16) = 29,844,570,422,669 p 312,500,000,000,000/17.90585 = 17,452,396,314,620.3125(16) = 279,238,341,033,925 p 3,125,000,000,000,000/19.05809 = 163,972,322,353,389.5625(16) = 2,623,557,157,654,233 p 31,250,000,000,000,000/20.21022 = 1,546,247,142,983,803.75(16) = 24,739,954,287,740,860 p 312,5(10¹⁴)/21.36225 = 14,628,604,204,771,537.9375(16) = 234,057,667,276,344,607 p 3,125(10¹⁵)/22.51421 = 138,801,225,160,057,427.5(16) = 2,220,819,602,560,918,840 p 31,25(10¹⁶)/23.66609 = 1,320,454,342,876,170,745.5(16) = 21,127,269,486,018,731,928 p 312,5(10¹⁷)/24.81792 = 12,591,705,418,082,244,143.125(16) = 201,467,286,689,315,906,290 p 3,125(10¹⁸)/25.96970 = 120,332,524,475,425,248,057.6875(16) = 1,925,320,391,606,803,968,923 p 31,25(10¹⁹)/27.1214392=1,152,224,985,459,325,054,241.625(16)=18,435,599,767,349,200,867,866 p 312,5(10²°)/28.273137 = 11,052,894,337,446,485,588,230(16) = 176,846,309,399,143,769,411,680 p 3,125(10²¹)/29.42480=106,202,921,929,527,321,332,975.1875(16)=1,699,246,750,872437141327603 31,25(10²²)/30.576438=1022028776677605027901712.4375(16)=16352460426841680446427399 p 312,5(10²³)/31.7280744=9849321314165923277849267.9375(16)=157589141026654772445588287 Knowing the evenly increasing ratio of primes to ½N from 10²⁸ by 1.151636, we can calculate that in the sequence of 50 (10²⁸) prime numbers there will be as many as the quotient of all numbers to half

20 of the given quantity, divided by the ratio in which there are prime numbers , which multiplied by 16 gives the exact number of prime numbers 3.125,(10²⁴)/32.879710 = 95.043.416.034.707.162.235.721,907.5(16) = 1.520.694.656.555.314.595.771.550.520, and in the sequence of 50 (10²⁹) prime numbers there will be / 32.879710 + 1.151636 = 34.031346 / and by this ratio we divide the quotient of half of the given quantity to obtain the number of primes 31.25(10²⁵)/34.031346 = 918.271.055,161,908,670,905,934.781.5(16) = 14,692,336,882,590,538,734,494,956,502.

RATIO OF TWIN NUMBERS TO PRIMES Twin numbers are two primes that can be represented as (6n ± 1), 6 (1) - 1 = 5, 6 (1) + 1 = 7, 6 (2) - 1 = 11, 6 (2) + 1 = 13. Here's how primes come together in twin pairs.

We can see that the number of the place on which they rank is a number whose 6 (n) ± 1 product creates them 6(10) - 1 = 59, 6(10) + 1 = 61, 6(12) - 1 = 71, 6(12) + 1 = 73. To 100 we have among 50 numbers there are seven pairs of twin numbers, i.e. 14 primes form pairs of twin numbers of the form p and p + 2. Because the numbers 2 and 3 as consecutive never form a twin pair, and 16 products of the number 3 cannot be counted as a twin number, because the number 5 cannot be paired with it and with 7 at the same time, and this would be counted twice in two pairs, i.e. for 100 we would have eight pairs and 15 twins, and this is not truth. The whole structure of 50 numbers can be put together in only 16 pairs, 9 of which are mixed pairs with products of numbers greater than 3 (23 - 25, 35 - 37, 47 - 49, 53 - 55, 65 - 67, 77 - 79, 83 - 85 , 89 - 91, 95 - 97). Subtracting from 25 - 2 = 23 the primes two numbers we get 23, of which we have to subtract 9 products of numbers greater than 3 (23 - 9 = 14), so only 14 prime numbers make 7 pairs of twins.

To ten there are 2 twins (5 - 7) to 4 primes (4 - 2 = 2), to 100 we have 25 - 14 = 11 primes that are not more paired, and 14 - 2 = 12 prime numbers, i.e. (11 - 2 = 9) nine unpaired primes, as well as 9 products of numbers greater than 3. The interdependence between these values is such that the difference in the number of primes and twins between the two intervals [π(N ') - π(N)] - [∑p,(p '+ 2) -

21 â&#x2C6;&#x2018;p,(p + 2)], (25 - 4 = 21) - (14 - 2 = 12), (21 - 12 = 9) , and also the amount of 9 products of numbers greater than 3 and 12 twins added together (9 + 9 + 12 = 30) must increase exponentially 3(q).

143 - 54 = 89, 166 - 9 = 157, 68 - 14 = 54, 157 + 89 + 54 = 300 = 30 (10) Up to 1000 among 334 prime numbers and their products greater than 3, there are twins 2(34) = 68. So 168 - 68 = 100 prime numbers does not form a twin numbers as in the range of 100 there were 25 - 14 = 11, the number of primes numbers do not form the twin increased by 100 - 11 = 89, and the products of more than 3 (166 - 9 = 157), and the twin numbers by (68 - 14 = 54). Adding these values together (157 + 89 + 54 = 300) we see that the number of primes and products greater than 3 involved in the formation of the twin numbers is constantly increasing exponentially 3(q) 30 - 300.

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In the line graph above, we can see exactly how the primes and twins are arranged in 16 sequences with constant spacing n(30), 11-13, 41-43, 71-73, 101-103, 29-31, 59-61, 149 151, ..

This property can be used when calculating the number of tins to a given magnitude. If the number of prime and twin numbers can be written as the quotient of the number 16, e.g. π (1000) = 16(10.5) = 168, and π₂(1000) = 16(4.25) = 68, it is easy to calculate the ratio of the numbers prime to tinsel 168/68 = 2.4705882 = 10.5/4.25.

Knowing the number of prime numbers written in the form of the quotient of the number 16 and the ratio in which there are twins to them, then dividing this quotient by the ratio we get the quotient of the twin numbers resulting from this ratio. It is worth noting that this ratio from π(10¹⁸)/π₂(10¹⁸) = 24,739,954,287,740,860/1,617,351,777,154,870 = 15,2965821, keeps increasing by 0.8730871. Thus, if there are 14,475,105,481,832,900 twins to π₂(10¹⁹) and 234,057,667,276,344,607 primes, their ratio is 16,1696692, and the next greater by 0.8730871 equals 17.0427563. So dividing this quotient of the number of prime numbers by the ratio 17.0427563 and multiplying by 16, (2,220,819,602,560,918,840)/16 = 138,801,225,160,057,427.5/17.0427563 = 8,144,294,427,307,948.25(16) = 130,308,710,836,927,172 we get the ratio of primes 17.0427,172 to the ratio of the prime numbers in this sequence. In this way, we can calculate any number of twins from the ratio.

21,127,269,486,018,731,928/16 = 1,320,454,342,876,170,745.5/17.9158434 73,703,163,920,051,385.625(16) = 1,179,250,622,720,822,170 p, p+2

=

23

24

STRUCTURES OF THE PRIME NUMBERS Mathematics is the study of structures if by structure we mean the way of joining individual parts together; constant, correct arrangement of components characteristic for a given object; structure, layout, style.

Here it may appear that mathematical objects acquire their identity in relation to their "relationship" with other objects. Thus, "numbers, points, straight lines, and planes" are defined by relational terms such as "in between" "among" "parallel" "congruent" and "continuous". This entire package then takes the form of a structure (or system). Thus, numbers, points, straight lines, and planes have relationships with each other, congruence, and complement each other in a certain proportion, and are parallel to other things because they are part of the whole structure (or system) in which these relationships take place. So what basic structure prime numbers form, here it is.

25 This structure is very simple. It consists of a left-handed double sequence of primes and their products greater than 3, constantly increasing by 6 (5-11-17-23-29), and to the right constantly by 8 (29-37-45-53-61), and straight ahead steadily by 2(7) = 14 (5-19-33-47-61-89).

The most interesting, however, is the relationship between prime numbers and their products in such a small group of numbers. Well, up to 96 there are 24 prime numbers and 24 their products (15 products of 3 and 9 products greater than 3), which gives 48 numbers, i.e. half of the given quantity (24 + 24 = 48), i.e. the ratio of prime numbers to their products is 1 : 1 according to the formula Ď&#x20AC;(N) + â&#x2C6;&#x2018;p(p') = ½N. The same ratio holds for 100, where for 25 primes we have 16 products of 3 plus 9 products of numbers greater than 3, i.e. 25 products (25 + 25 = 50) in the ratio 1 : 1.

Two facts are decisive when it comes to the arrangement of the prime numbers, which I hope to convince you of so much that you will remember it forever. The first is that prime numbers, despite their simple definition and role as building blocks of natural numbers, are building blocks for themselves, i.e. every prime greater than 3 is the sum of its predecessors, i.e. six primes (2 + 3 + 5 + 11 + 13 + 29 = 63 = 3(3)(7), n - this multiple of the prime number 7, [2, 3, 5, 11, 13, 29] + n(7) = p.

2=2 3=3 2+3=5 5+2=7 2(2) + 7 = 11 2(3) + 7 = 13 3 + 2(7) = 17 5 + 2(7) = 19 2 + 3(7) = 23 2 + 15(7) = 107 11 + 14(7) = 109 3 + 148(7) = 1 039

4(2) + 3(7) = 29 5 + 8(7) = 61 3 + 4(7) = 31 11 + 8(7) = 67 2 + 5(7) = 37 29 + 6(7) = 71 13 + 4(7) = 41 3 + 10(7) = 73 29 + 2(7) = 43 2 + 11(7) = 79 5 + 6(7) = 47 13 + 10(7) = 83 11 + 6(7) = 53 5 + 12(7) = 89 3 + 8(7) = 59 13 + 12(7) = 97 3 + 14(7) = 101 5 + 14(7) = 103 13 + 18(7) = 139 29 + 12(7) = 113 29 + 1430(7) = 10 039 5 + 142 862(7) = 1 000 039 5 + 142 857 142 862(7) = 1 000 000 000 039 3 + 1(7) = 10

2 + 14(7)

=

100

6 + 142(7)

=

1 000

4 + 1 428(7)

=

10 000

5 + 14 285(7)

=

100 000

1 + 142 857(7)

=

1,00E+06

3 + 1 428 571(7)

=

`1,00E+07

2 + 14 285 714(7)

=

1,00E+08

6 + 142 857 142(7)

=

1,00E+09

4 + 1 428 571 428(7)

=

1,00E+10

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5 + 14 285 714 285(7)

=

1,00E+11

1 + 142 857 142 857(7)

=

1,00E+12

3 + 1 428 571 428 571(7)

=

1,00E+13

2 + 14 285 714 285 714(7)

=

1,00E+14

6 + 142 857 142 857 142(7)

=

1,00E+15

4 + 1 428 571 428 571 428(7)

=

1,00E+16

5 + 14 285 714 285 714 285(7)

=

1,00E+17

4 + 1,428 571 428e99(7)

=

1,00E+100

4 + 1,428 571 428e999(7)

=

1,00E+1000

4 + 1,428 571 428e99 999 999(7)

=

1,00E+100 000 000

4 + 1,428 571 428e999 999 999(7)

=

1,00E+1000 000 000

The second fact is even more surprising in that it says that prime numbers are extremely regularly spaced and obey the law of assembly according to module 7 with extraordinary accuracy. Since all natural numbers (1,2,3,4,5,6,7,8,9,0) are consistent according to module 7, as shown in the diagram below, there are also prime numbers.

All the beauty of this structure is visible both in large and small three-dimensional spaces, where the formula p - (3,5,37,11,13,43,23,53,83,29,31,61) = 2n/7 is true

27

28

In such a structure, the prime numbers automatically distinguish themselves from their products in that they form 8 clockwise sequences with a constant spacing of 5(6) = 30, as shown below.

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In another structure, we see these spirally expanding prime numbers forming 12 double right-handed vortices with constant spacing 72, and their products left-handed.

30 In this structure, up to 1080, there are evenly 180 primes, products of numbers greater than 3, and blanks for products of 3, between spaces (7-4-11, 13-4-17)which, with a constant spacing of 70 to the left and 72 to the right, create these wonderful proportions;

25 (p) + 9 p(p')>3 = 34, 168 (p) + 166 p(p')>3 = 334, 334 - 34 = 300 30(p) + 30 ∑p(p') = 60 (½N), π(N)/∑p(p') = 1 : 1, π(N)/½N = 1 : 2 180 (p) + 180 ∑p(p')>3 + 180 ∑(3p) = 540 (½N), π(N)/∑p(p')>3 = 1 : 1, π(N)/½N = 1 : 3

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STRONG AND WEAK GOLDBACH HYPOTHESIS Now we are able to revisit the "strong" Goldbach hypothesis that every even natural number greater than 4 is the sum of two primes.

If the aspect ratio for all even numbers up to a given quantity is ½, then the equation ½N/N = π(N)/Σ(p + p') is the answer to Goldbach's problem that every even number can be composed of two prime numbers.

Claim: If the quotient of the number of prime numbers by their double number is equal to the quotient of the number of even numbers by a given quantity, then the two ratios are equal, i.e. the product of the extreme terms is equal to the product of the middle terms.

2k/N = 2k/Σ(p + p’) 50/100 = 50/(50 + 50) = ½

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The sum of two numbers with the same parity is always an even number /2k = p + p'/, as can be seen from the parity property. Hence, any even number greater than 4 can be represented as the sum of two even or prime numbers. /6 = 2 + 4 = 3 + 3, 8 = 2 + 6 = 3 + 5, 12 = 4 + 8 = 5 + 7, 14 = 2 + 12 = 3 + 11 = 6 + 8 = 7 + 7/

The proportion ½ for even numbers means that all even numbers in a given block consist of two prime numbers. 5/10 = 4/8, 50/100 = 25/50, 500/1000 = 168/336 There are 5 prime pairs up to 10 that add up to an even number 2 + 2 = 4, 3 + 3 = 6, 3 + 5 = 8, 5 + 5 = 10, 3 + 7 = 10, and choosing the ones closest to the middle to 100, we find 50 pairs of primes with an

33

even sum / 5 + 7 = 12, 3 + 11 = 14, 5 + 11 = 16, 7 + 11 = 18, 7 + 13 = 20, 5 + 17 = 22, 11 + 13 = 24, 7 + 19 = 26, 11 + 17 = 28/. So 50 even numbers in a block of 100 numbers is the sum of 2(25 + 25) 100 primes and the number of even-sum prime pairs grows exponentially by 5-50500-5000, with the common quotient q = 10, to infinity.

Thus, every even number greater than 4 may consist of 1 to 4 pairs of prime components, and yet there will be plenty of primes in a given block. 8 = 5 + 3, 10 = 7 + 3 = 5 + 5, 22 = 19 + 3 = 17 + 5 = 11 + 11, 36 = 19 + 17 = 23 + 13 = 29 + 7 = 31 + 5. Regardless how many prime numbers are in the range of numbers to a given magnitude, the even number there is always the sum of pairs of components of the preceding numbers, among which there will never be

34

no primes, which grow exponentially with their products 5-50-500, that is, as even-sum prime pairs.

The easiest way to find pairs of primes is by subtracting and adding the same number to half an even number, e.g. 105 (2516/2 = 1258 - 105 = 1153/1, 1258 + 105 = 1363/1, 1153 + 1363 = 2516) The validity of the "strong" Goldbach hypothesis proves the validity of the "weak" Goldbach hypothesis, because it is enough to subtract 3 from a given odd number greater than 7 and

35

present the resulting even number according to the strong Goldbach hypothesis. (2k + 1) - 3 = 2k = p + p 'â&#x2020;&#x2019; 2k + 1 = p + p' + p "

Now we can see that Goldbach's weak hypothesis holds true for all odd numbers, i.e. all odd numbers greater than 7 are the sum of three primes (not necessarily different), as seen in the graph above.

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Simply the proportional arrangement of the primes allows the sums of two primes / these numbers, when added in pairs, form a set of even integers / and sums of three primes / these numbers, when added with threes, form a set of odd integers / fill the number line with all natural numbers (except 1). In this simplest way, by pairing and triples, prime numbers can generate a set of natural numbers from apparent chaos. 2, 3, (2 + 2), (2 + 3), (3 + 3), (2 + 2 + 3), (3 + 5), (3 + 3 + 3), (5 + 5), (3 + 3 + 5), (5 + 7), (3 + 5 + 5), (7 + 7), (3 + 5 + 7), ..

Thus, from the apparent disorder of prime numbers, the extraordinary beauty of the proportion of their parts to other parts and to the whole set of natural numbers emerges, generating the most wonderful harmony with the human being, and according to Wisdom 11, 20 we can exclaim: "But you have precisely defined everything by measure, number and weight."

The apparent disorder is settled, for which God may be, thanks that we do not have to wait at least a million years for the understanding of the secrets of prime numbers.

Q E D "AD MAJOREM DEI GLORIAM - FOR THE GREATER GLORY OF GOD!"

37 TABLE OF PRIMES AND THEIR PRODUCTS GREATER THAN 3, GROWING IN THE PROGESSION OF 3(q)

38

39 TABLE OF PRIMES, TWINS AND THEIR PRODUCTS GREATER THAN 3 GROWING IN THE PROGESSION OF 3(q)

40