The prism of cognition

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2 ABSTRACT In the article "The Prism of Cognition" I would like to raise the issue of the existence of mathematical objects, especially such as prime numbers. Look at them through the prism of our previous knowledge about them and draw far-reaching conclusions from it. So let's go on the path of knowledge.

“God made the integers, all else is the work of man.� Leopold Kronecker WHAT WE KNOW ABOUT THE PRIME NUMBERS I don't know if the number 1,378,565,437 is a prime number, but I know it doesn't depend on me.

This is a foregone conclusion when we write this number, just like when we "build" a right triangle, we have a ^ + b ^ = c ^, whether we like it or not. Whether a given number is prime or not is determined by the properties it has on its own, i.e. whether it has only 2 factors, one and itself, or more factors, and then it is composite.

The basic property of prime numbers that distinguishes them from composite numbers is divisibility by 1 and itself. Each prime number consists of pairs of relatively prime components whose greatest common divisor is one (1 | [s + s']), hence they are not divisible by all other numbers and this fact is the best proof that a given number is a prime number. 11 = (10 + 1)/1,= (9 + 2)/1,= (8 + 3)/1,= (7 + 4)/1,= (6 + 5)/1, 5(11) = 55 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55/5 = 11/1, 11/11 = 1.

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 it is composite, 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 half 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 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

3 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

4 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, as we can see 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 into 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 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), lub 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 by this number we multiply it 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 prime 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

5 "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. For a long time these were purely theoretical questions, 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 for the compilation of the prime numbers, it may be time to approach the matter with a new strategy. So far the prime numbers have seemed to appear quite by chance. Such an attitude does not, of course, make it possible to predict what the prime number after 10,000 will be.

In fact, the distribution of prime numbers depends on a strict relation to its products, and this results from the ability to complete the primes by their products to half a given quantity, according to the formula Ď&#x20AC;(N) + â&#x2C6;&#x2018;p(p') = ½N, which says that the sum of the number of primes and their products equals half of the given quantity.

There are 80 numbers from 20 to 100, half of which are 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.

6 Up to 102/2 = 51 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.

Up to 102 there can be only 96/6 = 16 products of 3, to 1.020 is 1.014/6 = 169, and up to 10.200 is 10.194/6 = 1.699 products of number 3. So these are constants that make the sum of the prime numbers and products greater than 3 have the form (34 +1) = 26 + 9, (340 + 1) = 171 + 170, (3400 + 1) = 1252 + 2149. This obviously has a decisive influence on the ratio of prime numbers to products, which is shaped by as follows: 26/25, (1: 1) ± 0.5, 16 + 9 = 25, 171/339, 169 + 170 = 339, (1: 2) ± 1, 1.252/3.848, (1: 3) ± 23 , 1.699 + 2.149 = 3.848.

Even a cursory look at the sequence of numbers shows that odd numbers occupy a constant every second place, while the products of 3, every sixth place, and products of 5, every 30th place, and 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. It is this formula that determines the distribution of the prime numbers [π (N ) + ∑p (p ')> 3] + ∑3 (p) = ½N, (26 + 9) + 16 = (34 + 1) + (17-1) = 51.

The formula π(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 the products of the number 3 form a constant sum 17 - 1, increasing exponentially (17q) - 1. These two sums (34 + 1) + (17-1) = 51 complete each other half of the given quantity, which grows exponentially (51q). Hence we write [π (N) + ∑p (p ')> 3] + ∑3 (p) = ½N, (26 + 9) + 16 = (34 + 1) + (17-1) = 51 and that is the basic formula for the arrangement of prime numbers. We can see that the ratio of prime numbers to half of a given quantity 51/26 is equal to 2: 1 (- 0.5).

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 random tossing a coin or the "God does not play dice with the world" dice, but based on the ability to create identical intermediate sums of 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 numbers that follow regularly every 2 (3, 5, 7, 9, 11, 13, 15, 17, 19), and without Eratosthenes' sieve we have all selected primes and their products. At first glance, you can see with what regularity there are products of 3, every 18 numbers (9-27-45) or products of 5, every 30 numbers (25-55-85, 35-65-95) or products of 7, which 42 numbers (49-91-133), or products of 11, every 22 numbers (121-143, 187-209), or products of 13, every 78 numbers (169-247, 403-481), or 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 (11-29-47-65-83-101, 13-31-49-67-85-103), next to the series of products of 3 (15-33-51) and in these sequences the prime numbers are supplemented by their products to half the given quantity with the exemplary precision of 17 (p) + 13p (p ') = 30.34 (p) + 36p (p') = 70, 68 (p) + 102p (p ') = 170, ..

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It is difficult to imagine a more even distribution of prime numbers and their products than those resulting from the fact that they follow one another at constant distances, every 6 numbers, complementing each other in a strictly defined ratio (34 + 1) + (17-1) to half of a given quantity ½N = 51, as we can see in the line graph below. [π(N) + Σp(p ’)> 3] + ∑3 (p) = ½N. Therefore, although in the Riemann hypothesis, the prime distribution function π(x) is a gradual function of small serious irregularities, in the triple arithmetic sequence of primes and their products, with a constant interval D = 6, we see surprising smoothness. The uniformity with which this graph grows is not due to the

8 number of primes up to a given 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 arithmetic sequence of primes and their products .

We notice a similar regularity when we rank prime numbers according to the doubly characteristic numbers of ones 7 - 11 - 13 - 17 - 19 - 23 - 29 - 31. Then they form 8 sequences of prime numbers with a spacing n (30) = n (5 * 6) .

9 And here is the table of prime numbers and their products greater than 3, π(N) + ∑p(p')> 3 expanding exponentially (34q) + 1, and the products of the number 3, ∑3(p) in progress (17q) - 1.

For comparison, below is the function of the number of primes π(x) in the decimal system. You can see that these numbers 10/4, 100/25, 1,000/168, .. nothing to do with each other, because there is no visible relationship here.

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In contrast to the table below, showing how the prime numbers are completed to ½N.

11 Until now, the prime numbers have seemed quite randomly positioned on the number line. It has been observed that there are fewer primes, the larger the numbers we consider. When 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 π(N) + ∑p(p′) = ½N, i.e. they are mutually dependent. The prime numbers are also subject to the congruence law according to the modulus a ≡ b (mod 102), hence the number of primes for half a given quantity decreases asymptotically, while the number of their products increases asymptotically.

(½N)/∑p(p’) = (51/25)/25 = (2:1)+1, (510/339)/170 = (3:2)-1, (5100/3848)/170 = (30:22)+108

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To the number (N) 1020 we have 510 primes and their products in a triple arithmetic sequence, so there can be 510/3 = 170 numbers in one. In fact, there are 171 primes, but their products by one less are 340 - 1 = 339. We can calculate it from the formula [(½N)/2,3,4, .. - π (x)]/2 = ↕ π(N), ∑p(p'), that is half the difference between the third quotient of half a given quantity and the function of the number of primes π (x) in the decimal system [(510)/3 - 168]/2 = 2/2 = 1 and this one is added to the third quotient (510)/3 = (170 + 1) = 171, and we subtract from the double product 2(170) - 1 = 339 and this shows how the number of prime numbers affects their products - how many there are fewer primes, the more their products are. Up to 10,200 that is [(5100)/4 - 1229]/2 = (1275 - 1229)/2 = 46/2 = 23, it will be according to the 4th quotient by (1275 - 23) = 1252 fewer primes, and their products 3(1275) + 23 = 3825 + 23 = 3848 more.

171-number-long sequence of prime numbers from 2 - 1019, completed by 169 products of number 3 (9 - 1017), and 170 products of numbers greater than 3 (25 - 1015), [171 + (169 + 170)] = 171 + 339 = 510, to half the given quantity. From this layout it is clear that the products of the number three,

13 growing like a half of a given quantity in a geometric progression (17q) - 1 [169 = 17(10) - 1], have an influence on the number of prime numbers and their products greater than 3. That is, according to the basic ratio, there are 16 products of the number 3 with 34 + 1 = 35, i.e. 26 primes plus 9 of their products greater than 3. 16 + 35 = 51 = 26 + (16 + 6 +3) = 26 + 25, and for 169 products of 3 there are 171 primes and 170 products> 3. Prime numbers can be arranged in 16 sequences with a constant spacing n (30).

Or in 24 sequences increasing by n(48) and creating vortices with intervals n(50).

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And this is what it looks like on radar charts.

15 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, which proves that all numbers follow one rule that the sum of prime numbers and their products makes half of a given quantity π(N) + ∑p(p') = ½N, and the congruence law according to module 102. (161 – 59 = 102), a ≡ b (mod 102).

This is how 177 (p) + 176 (i> 3) + 175 (i3) = 528 (2) = among 1056 numbers is arranged.

16 When it comes to finding patterns and order, prime numbers are no longer an unsurpassed challenge. Knowing at what interval the next prime number or their product will appear, we can easily compile the entire list. And when we also have hints on how to determine the next number in the sequence, whether it is prime or complex, the list of primes does not appear to us as chaotic and random.

SUMMARY 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 an ordered, evenly increasing sequence of primes and their products up to half a given quantity. So the sum of prime numbers and their products greater than 3 π(N) + ∑p (p')> 3 equals the difference between a half of a given quantity ½N and the products of the number 3, ½N - ∑p(3), 171 + 170 = 341 = 510 - 169. The ratio of prime numbers to their products is then determined by the complement to half of the given quantity π(N) + ∑p(p') = ½N 1252 + 3848 = 5100, and the half of the difference between the products and primes [∑p(p') - π(N)]/2 (3848 - 1252)/2 = 1298 determines how many prime numbers are fewer and products more than ¼ of a given quantity (10200)/4 = 2550 - 1298 = 1252, 2550 + 1298 = 3848.

Finally, the mysterious structure of primes, tins and their products, searched for centuries by mathematicians, has been discovered and its music can be written endlessly.

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BIBLIOGRAPHY Sierpiński Wacław Elementary Theory of Numbers, Ziegler Günter The great prime number record races, Pomerance Carl The Search for Prime Numbers, Peter Bundschuh: Einführung in die Zahlentheorie, Paulo Ribenboim: The New Book of Prime Number Records, Narkiewicz Władysław The Development of Prime Number.

18 TABLES OF PRIMES AND THEIR PRODUCTS FROM 2 TO 1,023

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21 TABLES FOR PRIME AND TWIN NUMBERS FROM 2 TO 10,993

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