The key to prime numbers by Lubina

lubinaj7@gmail.com

“To this day, however, there is a widespread view bordering on certainty that prime numbers are arranged chaotically on the number line - there are no laws that would allow us to describe them all." INTRODUCTION 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 This combination of randomness and regularity motivated me to look for patterns in the arrangement of prime numbers that could eventually shed light on their ultimate nature. The discovery of these laws governing prime numbers gave me the key to getting to know them better. I want to share this knowledge now, and I hope that all math lovers will feel happy when they read these pages. PROPERTIES OF THE PRIME NUMBERS The basic property of prime numbers that distinguishes them from composite numbers is divisibility by 1 and itself. Each prime number is made up 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. Realizing that adding pairs of words from opposite ends of the list of numbers preceding a given odd number always yields identical intermediate sums (6 + 5) = 11 = (7 + 4), tells us whether the given triangular number as the sum of the numbers preceding the given quantity , consists of only 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 complex.

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

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 we can see in the table above, is a good example of this testing algorithm: t = [(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 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 complex 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

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 [π(N) + ∑p(p')> 3] + ∑3(p) = ½N, (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.

"Prime numbers are famous for creating an impenetrable tangle. According to many mathematicians, their order does not follow a discernible pattern". Vine Guy RATIO OF PRIME NUMBERS TO THEIR PRODUCTS Better understanding of prime numbers gives a mathematician the hope of finding 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 their way into 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 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. In fact, the distribution of prime numbers depends on a strict relation to their products, and this results from the ability to create identical intermediate sums of n - the number of pairs of extreme number components preceding a given quantity.

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, growing exponentially (17q)-1. These two sums (34+1) + (17-1) = 51 make up 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 one half of the given quantity 51/26 is as 2:1 ± 0.5.

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 above. [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 plot grows is not due to the 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 prime numbers and their products. 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.

Knowing these basic constant values, such as half of a given quantity ½N (51, 510, 5100), its three times smaller quantity N/3 = (34q + 1), containing prime numbers with their products

greater than 3, and a constant number of products of the number 3 equal to N/6 = (17q-1), we can easily calculate the number of primes with one precision. By subtracting from the sum of the products of prime numbers the known six times smaller number of products of numbers greater than 3, we get the number of products of 3 ∑p(p) ∑p(p')> 3 = ∑3(p), 339 - (1020/6) = 169. The number of products of numbers greater than 3, subtracted from the sum of the primes and their products greater than 3, gives us the exact number of primes to half of a given quantity. N/3 - ∑p(p')> 3 = π(N), 341 - 170 = 171, 171 + 339 = 1020/2 = 510. With the same accuracy as one, you can calculate the number of primes using their constantly increasing ratio from 1.01 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(p') = n/34, 9.766 + 41.234 = 51.000/34 = 1500, when 9,766 primes are paired with 41,234 their products, making a sequence of 34(1,500) = 51,000 numbers. If we now compare the quotient of the sum of the prime numbers and their products with the quotient of the prime numbers themselves 9.766/34 = 287.23529, we get 287.23529/1.500 = 1/5.2221995 what is the ratio of prime numbers to half of a given quantity. Conversely, dividing the quotient of all numbers by half of a given quantity by the ratio of prime numbers 1.500/5.2221995 = 287.23529, we obtain the quotient of prime numbers in this sequence, which multiplied by 34(287.23529) = 9.765.9998 gives the number of primes in the sequence with an accuracy of one tenth or 0.002. 15,000/6,23 = 2,404.3823(34) = 81,749 p 150,000/7.32 = 20,487.5(34) = 696,575 p 1,500,000/8.40 = 178,477.29(34) = 6,068,228 p 15,000,000/9.48 = 1,581,091(34) = 53,757,100 p 150,000,000/10.56 = 14,191,948.7(34) = 482,526,256 p 1,500,000,000/11.65 = 128,741,447.7(34) = 4,377,209,225 p 15,000,000,000/12.73 = 1,178,057,529(34) = 40,053,956,009 p 150,000,000,000/13.81 = 10,858,429,840(34) = 369,186,614,574 p 1,500,000,000,000/14.89 = 100,702,924,906(34) = 3,423,899,446,829 p 15,000,000,000,000/15.97 = 938,890,741,509(34) = 31,922,285,211,335 p 150,000,000,000,000/17.05 = 8,793,946,191,675(34) = 298,994,170,516,963 p 1,500,000,000,000,000/18.13 = 82,699,369,965,503(34) = 2,811,778,578,827,116 p

THE NUMBER OF PRIMES FUNCTION π(N) Until now, the prime numbers seemed to be arranged quite randomly 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. 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.

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

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 using the formula [(½N)/3 π(x)]/2 = ↕ π(N), ∑p(p'), i.e. 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 - o the fewer primes there are, the more of their products. 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. Interestingly, the number of quantities of their products ∑p(p') is a number always with the same parity as the number of prime numbers π(N). (171 + 339 = 510, 1252 + 3848 = 5100, 9766 + 41234 = 51000, 81749 + 428251 = 510000, ....) 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 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(N), then half of their sum and the difference added when their products are more than primes, or subtracted when the numbers there are fewer prime products than their products gives the exact value of π(N) to half of the given quantity. Proof: Up to 1020 we have 171 primes and 339 their products. Half of the sum and the difference of numbers with the same parity summed when their products are more than primes [339 + 171]/2 + [339 - 171]/2 = 510/2 + 168/2 = 255 + 84 = 339, or subtracted when there are fewer primes than their products [339 + 171]/2 - [339 - 171]/2 = 510/2 – 168/2 = 255 - 84 = 171, gives an exact value of π (N) to half the given quantity. 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, half the size. From this layout it is clear that the products of the number three, growing like half a given quantity in a geometric progression (17q) - 1 [169 = 17(10) - 1], affect the number of primes 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, that is, 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. 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 tips 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. This can be a list of prime numbers and their products which always complement them to a half of a given quantity π (N) + ∑p (p ') = ½N, for 1.020 we

have 171 primes plus 339 of their products, which gives a total of 510 all numbers, which are half a given magnitude of 1,020.

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.

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 â&#x2030;Ą b (mod 102).

Here we see how spirally developing prime numbers and their products greater than 3 with constant spacing (2) - (4) = (6) (17 - 19 - 23 - 25 - 29) form 34 columns of numbers adjacent to each other in a constant d-spacing 102 = 6(17) (25 - 127 - 229) and on the radar chart they arrange themselves into 17 left-handed vortices with a constant distance d -(50) - (52) = 102. MYSTERIES OF TWIN NUMBERS Twin numbers are two prime numbers that can be represented in the form (6n±1), 6(1)-1 = 5, 6(1)+ 1 = 7, 6(2) - 1 = 11, 6(2) + 1 = 13. And 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 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 π₂ (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 {π(N) + ∑p(p') > 3]/34 = π₂(N)/34 + d/34, 341/34 = 10(34) + 1 = [69/34 + 1] + 272/34 = 8(34)] + [2(34)+1] = 10(34)+1 This property can be used when calculating the number of twins to a given magnitude. If the number of primes and twins can be written as the product of 34, for example: π(1020) = 5 (34) + 1 = 171, and π₂(1020) = 2(34) + 1 = 69, then it is easy to calculate what is the ratio of twins to primes 69/171 = 2:5 + 1. In the interval 10000≤p≤10200, there are 23 prime numbers, i.e. (1229 + 23) = 1252 primes, and the twin numbers 10, i.e. a sum of π₂ (10000) + π₂ (10200) = 408 + 10 = 418 numbers

10007, 10009, 10037, 10039, 10061, 10067, 10069, 10079, 10091, 10093, 10099, 10103, 10111, 10133, 10139, 10141, 10151, 10159, 10163, 10169, 10177, 10181, 10193. There are 174 prime numbers in the interval π(100000≤p≤102000), i.e. (9592 + 174) = 9766 in the sum of prime numbers and 29 twins π₂(100000) + π₂(102000) is (2447 + 29) = 2476 100003, 100019, 100043, 100049, 100153, 100169, 100183, 100189, 100279, 100291, 100297, 100313, 100391, 100393, 100403, 100411, 100501, 100511, 100517, 100519, 100609, 100613, 100621, 100649, 100741, 100747, 100769, 100787, 100853, 100907, 100913, 100927, 100999, 101009, 101021, 101027, 101113, 101117, 101119, 101141, 101203, 101207, 101209, 101221, 101323, 101333, 101341, 101347, 101419, 101429, 101449, 101467, 101527, 101531, 101533, 101537, 101627, 101641, 101653, 101663, 101741, 101747, 101749, 101771, 101863, 101869, 101873, 101879, 101963, 101977, 101987, 101999.

100057, 100193, 100333, 100417, 100523, 100669, 100799, 100931, 101051, 101149, 101267, 101359, 101477, 101561, 101681, 101789, 101891,

100069, 100207, 100343, 100447, 100537, 100673, 100801, 100937, 101063, 101159, 101273, 101363, 101483, 101573, 101693, 101797, 101917,

100103, 100213, 100357, 100459, 100547, 100693, 100811, 100943, 101081, 101161, 101279, 101377, 101489, 101581, 101701, 101807, 101921,

100109, 100237, 100361, 100469, 100549, 100699, 100823, 100957, 101089, 101173, 101281, 101383, 101501, 101599, 101719, 101833, 101929,

100129, 100267, 100363, 100483, 100559, 100703, 100829, 100981, 101107, 101183, 101287, 101399, 101503, 101603, 101723, 101837, 101939,

100151, 100271, 100379, 100493, 100591, 100733, 100847, 100987, 101111, 101197, 101293, 101411, 101513, 101611, 101737, 101839, 101957,

Knowing the number of prime numbers written in the form of the quotient of the number 34 and the ratio of the twin numbers to them, dividing this quotient by the ratio we obtain the quotient of the twin numbers resulting from this ratio. It is worth noting that this ratio, from π₂(10,200)/π(10,200) = 418/1252 = 1/3, is constantly increasing by 0.9. If there are 418 twins and 1252 primes to π₂(10,200), the ratio is 1/3 and the next 0.9 greater is 1/3.9. So dividing this quotient of the number of primes by the ratio 3.9 and multiplying by 34, (9766)/34 = 287/3.941 = 72.8235(34) = 2476, we get the number of twins in the ratio 3.9 to the primes in this sequence. π₂(N)/π(N) 2(34)+1/5(34)+1, 2/5 12.2(34)/36.8(34), 1/3 72(34)/287(34), 1/3.9 500.8(34)/2,404(34), 1/4.8 3,594(34)/20,487.5(34), 1/5.7 27,042(34)/178,477(34), 1/6.6 210,812(34)/1,581,091(34), 1/7.5 1,689,517.7(34)/14,191,948.7 (34), 1/8.4 13,843,166(34)/128,741,447.7 (34), 1/9.3

π₂(N) 69/34 = (2)+1 418/34 = 12.2 2,476/34 = 72 17,027/ 34 = 500.8 122 205/34 = 3 594 919 428/34 = 27 042 7,167,608/34 = 210 812 57,443,601/34 = 1,689,517.7 470,667,658/34 = 13,843,166

π(N) 171/34 =(5)+1 1,252/34 = 36.8 9,766/34 = 287 81,749/34 = 2,404 696 575/34 = 20 487.5 6,068,228/34 = 178,477 53,757,100/34 = 1,581,091 482,526,256/34 = 14,191,948.7 4,377,209,225/34 = 128,741,447.7

115,495,836(34)/1,178,057,529(34), 1/10.2

3,926,858,427/34 = 115,495,836

40,053,956,009/34 = 1,178,057,529

20 978236922(34)/10858429840(34), 1/11.1 8391910408(34)/100702924906(34), 1/12 72782228023(34)/938890741509(34), 1/12.9 637242477657.5(34)/ 8793946191675(34),1/13.8 5625807480646(34)/82699369965503(34),1/14.7 50031379733290(34)/780489523839326(34),1/15.6

33,260,055,348/34 = 978,236,922 285,324,953,872/34 = 8,391,910,408 2,474,595,752,815/34 = 72,782,228,023 21,666,244,240,355/34 = 637,242,477,657.5 191,277,454,341,977/34 = 5,625,807,480,646 1,701,066,910,931,860/34 = 50,031,379,733,290

369,186,614,574/34 = 10,858,429,840 3,423,899,446,829/34 = 100,702,924,906 31,922,285,211,335/34 = 938,890,741,509 298,994,170,516,963/34 = 8,793,946,191,675 2,811,778,578,827,116/34 = 82,699,369,965,503 26,536,643,810,537,096/34 = 780,489,523,839,326

As we can see, up to the number 360/2 = 180 we have 20 pairs, i.e. 40 twin numbers by a further 40 to 80 numbers will increase after 800 numbers with 1160/2 = 580, so that after 840 numbers with 2000/2 = 1000 there are 120, and after 980 numbers at 2980/2 = 1490 there are 40 more, or 160 twin numbers. So evenly distributed twin numbers are constantly 40 more (40-80-120-160-200-240-280-320-360-400,..).

SUMMARY Thus, the puzzle of the distribution of prime numbers has been solved. From now on, 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 of the 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 3, ½N - ∑p(3), 171 + 170 = 341 = 510 - 169. The ratio of prime numbers to their products is thus determined by the complement to half of a 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 primes are less and the products more than 1/4 of a given quantity (10200)/4 = 2550 - 1298 = 1252, 2550 + 1298 = 3848. Finally, the mysterious structure of primes, twins and their products, searched for centuries by mathematicians, has been discovered and its music can be written endlessly.

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.

TABLES FOR PRIME AND TWIN NUMBERS 2 TO 1,021

TABLES OF PRIME NUMBERS AND THEIR PRODUCTS FROM 2 TO 1,023

TABLES OF PRIMES AND TWIN NUMBERS FROM 2 TO 10,993