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The key to prime numbers

Profile image of Jan LubinaJan Lubina

2020, paper

Abstract

Prime numbers are of greater interest to mathematicians, both professional and amateur, since people began to study the properties of numbers and find them fascinating. On the one hand, prime numbers seem to be randomly distributed among natural numbers with no law other than probability. On the other hand, however, the distribution of primes globally reveals remarkably smooth regularity when viewed in the context of their products. This can be described by the formula π(N) + ∑p(p ’) = ½N, which says that half of a given quantity is the sum of the number of primes to a given quantity and their products. The combination of the number of prime numbers π(N) with their products greater than 3 ∑p(p')> 3 always creates a constant value growing in progress 34+1(q), and their products of the number 3 ∑3(p) in progress 17-1(q), and half of a given quantity of ½N progressively 51(q). (34 + 1)q + (17-1)q = (51)q, [π(N) + ∑p(p ')> 3] + ∑3(p) = ½N, (26+9) + 16 = (34+1)+(17-1) = 51

Figures (38)
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 complex number. The number 11 is prime because the five pairs of identical sums that make it up, added outright 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
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 complex number. The number 11 is prime because the five pairs of identical sums that make it up, added outright 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
Even a cursory look at the sequence of numbers shows that odd numbers occupy a constant every second place, while the products of the number 3, every sixth place, and the products of 5, every 30th place and the products of 7, every 42nd place. This obviously has a decisive influence on the number of primes and their products to a given quantity. Therefore, in 102 numbers out of 51 numbers there are 26 primes, 16 products of 3 (9,15,21, ...) 6 products of 5 (25,35,55,65,85.95), and 3 products of 7 ( 49.77.91). 26+ 16+ (6+ 3)=51=(26+9)+16= 35 + 16 = (34 + 1) + (17-1) = 51. The arrangement of the prime numbers depends on this formula [m(N) + >p(p')> 3] + ¥3(p) = “’N, (26 + 9) + 16 = (34 + 1) + (17-1) = 51. This shows the perfect order in the whole a sequence of natural numbers, consisting in 50% of even and odd numbers, i.e. prime numbers and their products. Such basic numbers are not determined by nature by the method of random tossing a coin or dice "God does not play dice with the world", but based on the ability to create identical intermediate sums of n - the number of pairs of extreme number components preceding a given quantity. Chance and chaos are simply unacceptable to mathematics. tis based on than 1. Then prime factors prime factors the fundamental property of prime numbers to form n - the number of pairs ot components with identical intermediate sums that do not have a common divisor greater the triangular number as the sum of all preceding numbers decomposes into up to the given number, which means that is prime. When it decomposes into ess than a given number, it is a complex number.
Even a cursory look at the sequence of numbers shows that odd numbers occupy a constant every second place, while the products of the number 3, every sixth place, and the products of 5, every 30th place and the products of 7, every 42nd place. This obviously has a decisive influence on the number of primes and their products to a given quantity. Therefore, in 102 numbers out of 51 numbers there are 26 primes, 16 products of 3 (9,15,21, ...) 6 products of 5 (25,35,55,65,85.95), and 3 products of 7 ( 49.77.91). 26+ 16+ (6+ 3)=51=(26+9)+16= 35 + 16 = (34 + 1) + (17-1) = 51. The arrangement of the prime numbers depends on this formula [m(N) + >p(p')> 3] + ¥3(p) = “’N, (26 + 9) + 16 = (34 + 1) + (17-1) = 51. This shows the perfect order in the whole a sequence of natural numbers, consisting in 50% of even and odd numbers, i.e. prime numbers and their products. Such basic numbers are not determined by nature by the method of random tossing a coin or dice "God does not play dice with the world", but based on the ability to create identical intermediate sums of n - the number of pairs of extreme number components preceding a given quantity. Chance and chaos are simply unacceptable to mathematics. tis based on than 1. Then prime factors prime factors the fundamental property of prime numbers to form n - the number of pairs ot components with identical intermediate sums that do not have a common divisor greater the triangular number as the sum of all preceding numbers decomposes into up to the given number, which means that is prime. When it decomposes into ess than a given number, it is a complex number.
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 for the compilation of a prime number list, it may be 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.
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 for the compilation of a prime number list, it may be 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.
It is difficult to imagine a more even distribution of prime numbers and their products thar those resulting from the fact that they follow one another at constant distances, every € numbers, complementing each other in a strictly defined ratio (34 + 1) + (17-1) to half of ; given quantity “4N = 51, as we can see in the line graph above. [m(N) + 2p(p’)> 3] + >3(p) = “%N. Therefore, although in the Riemann hypothesis, the prime distribution function m (x) is < gradual function of small serious irregularities, in the triple arithmetic sequence of prime: and their products, with a constant interval D = 6, we see surprising smoothness. The uniformity with which this plot grows is not due to the number of primes up to a giver quantity N, which can be localized by a logarithmic function, but to their regular distribution which comes from the constant difference d = 6 between the members of the triple The formula m(N) + ¥p(p')> 3 = 34q+1 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 34 + 1 growing exponentially (34q)+1. Also t he produc 1, growing exponentially (17q)-1. These two su given quantity, which grows exponen YN, (26 + 9) + 16 = (34+ 1) + (17-1 tially (51q prime numbers. We can see that t quantity 51/26 is as 2:1+0.5. ts of the number 3 form a constant sum 17 | ms (34+1) + (17-1) = 51 make up half of the . Hence we write [m(N) + }p(p')> 3] + >3(p) = 51 and that is the basic formula for the arrangement o he ratio of prime numbers to one half of the giver
It is difficult to imagine a more even distribution of prime numbers and their products thar those resulting from the fact that they follow one another at constant distances, every € numbers, complementing each other in a strictly defined ratio (34 + 1) + (17-1) to half of ; given quantity “4N = 51, as we can see in the line graph above. [m(N) + 2p(p’)> 3] + >3(p) = “%N. Therefore, although in the Riemann hypothesis, the prime distribution function m (x) is < gradual function of small serious irregularities, in the triple arithmetic sequence of prime: and their products, with a constant interval D = 6, we see surprising smoothness. The uniformity with which this plot grows is not due to the number of primes up to a giver quantity N, which can be localized by a logarithmic function, but to their regular distribution which comes from the constant difference d = 6 between the members of the triple The formula m(N) + ¥p(p')> 3 = 34q+1 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 34 + 1 growing exponentially (34q)+1. Also t he produc 1, growing exponentially (17q)-1. These two su given quantity, which grows exponen YN, (26 + 9) + 16 = (34+ 1) + (17-1 tially (51q prime numbers. We can see that t quantity 51/26 is as 2:1+0.5. ts of the number 3 form a constant sum 17 | ms (34+1) + (17-1) = 51 make up half of the . Hence we write [m(N) + }p(p')> 3] + >3(p) = 51 and that is the basic formula for the arrangement o he ratio of prime numbers to one half of the giver
Knowing these basic constant values, such as half of a given quantity “N (51, 510, 5100), its three times smaller quantity N/3 = (34q + 1), containing prime numbers with their products
Knowing these basic constant values, such as half of a given quantity “N (51, 510, 5100), its three times smaller quantity N/3 = (34q + 1), containing prime numbers with their products
Until now, the prime numbers seemed to be arranged quite randomly on the number line. | has been observed that there are fewer primes, the larger the numbers we consider. Wher it comes to their arrangement, prime numbers follow one rule that the sum of prime numbers and their products makes up half of a given quantity m(N) + >p(p') = “N, i.e. they are mutually dependent. Prime numbers are also subject to the congruence law of modulus 102, hence the number of primes for half a given quantity decreases asymptotically, while the number of their products increases asymptotically.
Until now, the prime numbers seemed to be arranged quite randomly on the number line. | has been observed that there are fewer primes, the larger the numbers we consider. Wher it comes to their arrangement, prime numbers follow one rule that the sum of prime numbers and their products makes up half of a given quantity m(N) + >p(p') = “N, i.e. they are mutually dependent. Prime numbers are also subject to the congruence law of modulus 102, hence the number of primes for half a given quantity decreases asymptotically, while the number of their products increases asymptotically.
The table shows the dependence of the number of products of prime numbers on the number of prime numbers themselves.
The table shows the dependence of the number of products of prime numbers on the number of prime numbers themselves.
The table shows by which number the number of primes decreases and the number of their products increases.
The table shows by which number the number of primes decreases and the number of their products increases.
By subtracting those 34 numbers (53, 103, 55, 5, 59, 7, 61, 11, 65, 13, 17, 67, 71, 19, 23.73, 77, 25, 79, 29, 83, 31, 85, 35, 89, 37, 91, 41, 95, 43, 97, 47, 101, 49), from the number following it in a straight line we always get a number divisible by 102 e.g. 457 - 49 = 408/102 = 4, what it is the proof that all figures are subject to a single law distribution, according to the law of congruence module 102 (161 - 59 = 102), a=b (mod 102). And this is what it looks like on radar charts.
By subtracting those 34 numbers (53, 103, 55, 5, 59, 7, 61, 11, 65, 13, 17, 67, 71, 19, 23.73, 77, 25, 79, 29, 83, 31, 85, 35, 89, 37, 91, 41, 95, 43, 97, 47, 101, 49), from the number following it in a straight line we always get a number divisible by 102 e.g. 457 - 49 = 408/102 = 4, what it is the proof that all figures are subject to a single law distribution, according to the law of congruence module 102 (161 - 59 = 102), a=b (mod 102). And this is what it looks like on radar charts.
We can see that the number of the place on which they rank is a number whose 6(n) + 1 product creates 6(10) - 1 = 59, 6(10) + 1 = 61. To 102 we have seven pairs of numbers among 35 numbers twins plus one 2(7) + 1 = 15. To 1020 of 341 primes and their products greater than 3, there are twins 2(34) + 1 = 69. Subtracting from 341 - 69 = 272 we get the number of numbers that do not form pairs of twin numbers. Interestingly, since the number rz (1020) = (341 - 69) = 272/34 = 8, all numbers are divisible by 34. This proves that all prime numbers and their products are arranged in order 3(34) = 102 and according to this measure, they are arranged in relation to each other. Hence we can write the sum of the number of prime numbers and their products greater than 3, divisible by 34, equal to the sum of the number of twins and the number of numbers that do not form pairs of twin numbers "d" divisible by 34 {n(N) + >p(p') > 3]/34 = m2(N)/34 + d/34, 341/34 = 10(34) + 1 = [69/34 + 1] + 272/34 = 8(34)] + [2(34)+1] = 10(34)+1
We can see that the number of the place on which they rank is a number whose 6(n) + 1 product creates 6(10) - 1 = 59, 6(10) + 1 = 61. To 102 we have seven pairs of numbers among 35 numbers twins plus one 2(7) + 1 = 15. To 1020 of 341 primes and their products greater than 3, there are twins 2(34) + 1 = 69. Subtracting from 341 - 69 = 272 we get the number of numbers that do not form pairs of twin numbers. Interestingly, since the number rz (1020) = (341 - 69) = 272/34 = 8, all numbers are divisible by 34. This proves that all prime numbers and their products are arranged in order 3(34) = 102 and according to this measure, they are arranged in relation to each other. Hence we can write the sum of the number of prime numbers and their products greater than 3, divisible by 34, equal to the sum of the number of twins and the number of numbers that do not form pairs of twin numbers "d" divisible by 34 {n(N) + >p(p') > 3]/34 = m2(N)/34 + d/34, 341/34 = 10(34) + 1 = [69/34 + 1] + 272/34 = 8(34)] + [2(34)+1] = 10(34)+1
twins in the ratio 3.945 to the primes in this sequence.
twins in the ratio 3.945 to the primes in this sequence.
Finally, the mysterious structure of primes, twins and their products, searched for centuries
Finally, the mysterious structure of primes, twins and their products, searched for centuries
TABLES OF PRIME NUMBERS AND THEIR PRODUCTS FROM 2 TO 1,023
TABLES OF PRIME NUMBERS AND THEIR PRODUCTS FROM 2 TO 1,023

Key takeaways

  • On the one hand, prime numbers seem to be randomly distributed among natural numbers with no law other than probability.
  • The discovery of these laws governing prime numbers gave me the key to getting to know them better.
  • This shows the perfect order in the whole a sequence of natural numbers, consisting in 50% of even and odd numbers, i.e. prime numbers and their products.
  • With the same accuracy as one, you can calculate the number of primes using their constantly increasing ratio from 1.08 to 1.14 of prime numbers to half of a given quantity This is because both prime numbers and their products form sequences, and the sum of the sequence is divisible by 34 π(N) + ∑p ( Until now, the prime numbers seemed to be arranged quite randomly on the number line.
  • (1229 + 23) = 1252 primes, and the twin numbers 10, i.e. a sum of π₂ (10000) + π₂ (10200) = 408 + 10 = 418 numbers SUMMARY Thus, the puzzle of the distribution of prime numbers has been solved.

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