2 ABSTRACT "In a maze of prime numbers" is study of properties of integers, based on the theories of arithmetic operations on integers and in my work; I tried to add some explanation to the basic model lying behind the primes. The article tells what exactly the prime number is? Which numbers are primes? Why is this number prime? Where do the prime numbers come from? Where are the primes going to? How are primes distributed, how to recognize and test them. It also points the interrelations between the primes and their products, which helps us to understand how more and more decreasing to a given magnitude quantity of primes is related to more and more increasing their products. Trying to explain what makes a given numbers to be prime numbers or its products. This paper explores some of the basic properties of prime numbers and several theorems associated with them. It explains how the primes are woven into an infinite sequence of natural numbers, and reveals all its hidden so far beauty, which is a reflection of eternally encoded in the order. All these give us to reach the proverbial net to capture in its other outstanding issues, such as these of distribution of primes and twin primes, their density and many others. "EVERYTHING IS A NUMBER" (Pythagoreans) THE CONCEPT OF NUMBERS Anyone, even for years not dealing with mathematics, can distinguish an even number, divisible by 2 because it is its product and all natural numbers 2 (1, 2, 3, 4, 5, ...) = 2n from the odd number, which are divided into prime numbers and their products, that is, complex numbers. If we analyze the definition of a prime number, we will quickly figure out which number deserves the name of the prime and which of the name of the complex. We know that any integer greater than one, divisible only by 1, and itself is prime. From this definition it follows 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)/1 = 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)/1 = 11/1, because they make up the prime number. 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
3
But not all odd numbers are prime numbers, and although they are formed on the same principle, when decoding them we will see that they are not only composed 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, 51 = (3 + 48)/3.
4 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 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 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 basic 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 confirming that the given number is a prime number or their product.
5
[(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 – 1)/2](n) = t | p = (n = p), lub t | p = (p < n) = p(p’) So 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 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
We already know what a prime number is exactly, how it arises and what it consists of. How to recognize it among other odd numbers and test it. So it's time to take a look at how they are develop.
6 "Numbers have a way of taking a man by the hand and leading him down the path of reason." – Pythagoras HOW PRIMARY NUMBERS DEVELOP
Better understanding of prime numbers implies the hope of a mathematician 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. The 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 proof of Riemann's conjecture can tell us anything about the distribution of prime numbers in the world of numbers. For centuries, the magic formula has been searched in vain, it may be time to come up with a list of prime numbers, so it's 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 complement prime numbers by their products to half of a given value, according to the formula π(x) + ∑ p(p') = ½N, which says that the sum of the number of primes and their products is equal to half of the given quantity. From 20 to 100 there are 80 numbers, half of which is 40 = 17 are prime numbers plus 23 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(x) + ∑ 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 (4 -30- 34 -300- 334), 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(x) + ∑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 2 : 1. This testifies to the perfect order that prevails over 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 dice roll "God does not play dice with the world", but based on the ability to create constant sums of prime numbers and their products greater than 3 (4 -30- 34 -300-334), exponentially growing 3 (q). Chance and chaos are simply unacceptable to mathematics. So it is enough to create a table of 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 number 5, every 30 numbers (25-55-85, 3565-95) or products of number 7, every 28 numbers (49-77, 133-161), or products of number 11,
7 every 88 numbers (121-209, 187-275), or products of 13, every 26 numbers (221-247, 377-403), or products of 17, every 34 numbers (289-323).
The radar chart below clearly shows that prime numbers together with products greater than 3 form two separate arithmetic sequences (5-11-17-23-29-35,.. 7-13-19-25-31), which parallel to the products of the number 3 (9-15-21-27) expand in a spiral every 6 (14) = 84 numbers, 5 -84- 89, 7 -8491, 9 -84-93 and in these 3 sequences the primes are supplemented by their products to half the given quantity with exemplary accuracy [10(p) + 5p(p')]/5 = 15/5, 2 : 1, [16 (p) + 12 p(p')]/4 = 28/4, 4 : 3, 30(p) + 30p(p') = 120/2 = 60, 1 : 1, ..
8 So there is a strict relationship between the number of arriving primes π(x) and their products greater than three /∑ [p(p')]> 3/, which increase exponentially 3(q), as can be seen in the table below. /3(q) = ∑[p(p')]> 3 + π(x), 30 = 9 + 21, 300 = 157 + 143,.. / That is, if there are 4 prime numbers in the top ten, then up to 100 there cannot be more than 3(10) = 30, i.e. 21 primes plus 9 of their products greater than 3, this equals 30. 4 + 30 = 34 + 300 = 334 + 3000 = 3334, .. ∑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 29 215 278 521
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 4 118 054 813
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 33 333 333 334
N 10 10² 10³ 10⁴ 10⁵ 10⁶ 10⁷ 10⁸ 10⁹ 10¹⁰ 10¹¹
This leads to a balanced distribution of prime numbers and their products greater than 3 included in 2 arithmetic series with constant spacing r = 6 and expanding exponentially 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(x) + ∑3p = ½N, always completed to ½ N by the products of 3 (16, 166, 1 666), as the sum of two constant numbers 34 + 16 = 50, 334 + 166 = 500, 3 334 + 1 666 = 5,000, and this 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(x)] + d [∑p(p')> 3] = 3(q ) = [π(x') + ∑p(p')> 3] - [π(x) + ∑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. 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 number of products greater than 3 to primes increases asymptotically, and the number of primes asymptotically decreases.
9
“Reach where the eyes cannot see; Break what your mind cannot break. Adam Mickiewicz PROPORTIONAL DEVELOPMENT OF THE FIRST NUMBERS Each natural number is either a prime number or a prime product, meaning that each integer uniquely decomposes prime number factors. 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 and how they develop. At the basis of the development of prime numbers in a sequence of natural numbers is their decomposition into adducts, and their products into prime factors. To 10 we have 4 primes composed of 8 adders: 2 = 1 + 1, 3 = 1 + 2, 5 = 2 + 3, 7 = 3 + 4 and the product of 3 (3) = 9 composed of 2 factors. Similar to 100, we have 25 primes composed of 50 adders and 25 products of 50 prime factors. Up to 1000, we have 168 primes of 336 adders and 332 products of 664 factors. Why 168 primes? Is there any rule or formula that can tell me how many primes are up to a given magnitude? Can you find the laws that describe their dilution? There is a function π (x) that reproduces the number of primes 2 up to a given quantity. But how can π(x) be calculated? Can we just write 5 : 10 = 4 : 8 and 50 : 100 = 25 : 50, 500 : 1000 = 168 : 336, y = ½. Another thing about this purely mathematical problem is that mathematicians focus too much on the prime numbers themselves, and ignore other and interrelationships between them. These are only partially recorded. But you cannot exclude whole groups of numbers from context and ignore other numbers. Now that we have learned more about the properties of prime numbers, let us now try to look at their development. You have noticed that after eliminating even numbers, which are half of a given quantity, the density of prime numbers in a set of numbers to a given quantity has the same as half of a given quantity to all given quantity ½N: N = 0.5. The formula is therefore simple and results from the equality of two ratios, when the quotient of half of a given quantity ½N/N by a given quantity is equal to the quotient of the number of prime numbers π(x)/∑(n+n') by the sum of their components.
10
11
12 In mathematics, two quantities are proportional to each other; when they change in this way that one quantity is always a multiple of the other, or, which is equivalent to being in constant relation to each other. Proportion, then, is the equality of relations. In proportional quantities, the multiplication or division of one quantity is constantly associated with the multiplication or division of the other, or generally speaking: one quantity results from the other by constantly multiplying the same ratio of both quantities, called the proportionality factor or the proportionality constant. Claim: If the quotient of a half of a given quantity ½N/N by a given quantity is equal to the quotient of the number of prime numbers π(x)/ ∑(n + n') by the sum of the components included in the prime numbers, then the two ratios are equal and then the product of the extreme terms is equal to the product of middle terms. Proof: ½N : N = π(x) : ∑(n + n’), π(x) ~ ½N, ½N * ∑(n + n’) = N * π(x), π(x)/∑(n + n’) = y = ½ 5/10 = 4/8 = 50/100 = 25/50 = 500/1000 = 168/336 = ½, 5 * 8 = 10 * 4, 50 * 50 = 100 * 25, 500 * 336 = 1000 * 168, ∑(n + n’) * 0.5 = π(x), 8 * 0.5 = 4, 50 * 0.5 = 25, 336 * 0.5 = 168 Thus, prime numbers are not completely randomly distributed among other numbers, but are subject to strict rules governing arithmetic sequences that expand proportionally to the entire given quantity. It has long been known that there are fewer primes the larger the numbers we consider because their number is inversely proportional to the number of numbers in a given quantity. Inverse proportionality occurs between two quantities, the product of which is constant in this case between half of a given quantity and the number of prime numbers ½N * ∑(n + n') = a, then π(x) = a * 1/N, 5 * 8 = 40, 4 = 40/10. Inversely proportional means that each quantity π (x) is directly proportional to the reciprocal of the product of half of a given quantity and the sum of its components by the given quantity. π(x) ~ ½N * [∑(n + n')]/N, π(x) = ½N * [∑(n + n')]/N, 4 = 5 *(8)/10, 25 = 50 * (50)/100, 168 = 500 * (336)/1000, 1229 = 5000 * (2458)/10000.
The graph of the proportion function shows that it is asymptotically decreasing, which means that there are fewer primes in a given quantity, the larger the numbers we consider. If in 100 numbers
13 out of 50 odd, every second, i.e. 25 is prime, then they are in the ratio of 25/50 = 0.5, then in 1000 numbers this ratio is 168/500, i.e. 0.336.
Looking at the table above, we see how the proportionality factor steadily declines from 0.8 through 0.5 to 0.029384673765181077468989913004 as it approaches zero, which causes ever greater dilution of the prime numbers in a given quantity, but not their disappearance. The line of the proportionality factor on the coordinate axis creates a hyperbola that begins with a product with factors 0.8 * 5 = 4, 0.5 * 50 = 25, 0.336 * 500 = 168 increasing exponentially 5, 50, 500, all the way to infinity is the best proof that prime numbers are inversely proportional to a given size there are infinitely many.
14
If half of a given quantity ½N, it consists in the proportion ½ of complex components and factors /1 + 4 = (2 + 8) = 10 * 0.5/ and the sums of all prime numbers are on the line lying halfway to the specified sum of prime numbers and their products , then considering the sums of primes in relation to a given quantity N, we see that their hyperbolas can approach the coordinate axis asymptotically by a distance of 0.5 * [2π(x)], /0.5*2(4) = 0.5 * 8 = 4/, /0.5*2(25) = 0.5 * 50 = 25/, 0.5 * 2(168) = 0.5 * 336 = 168, that is, all sums of prime numbers ∑(p) = 0.5 * [2π (x)] lie on the line 0.5.
15
The ½ proportion means that double the number of their adders and factors takes part in creating half of the numbers in a given block. The very distribution of prime numbers and therefore the Riemann Hypothesis says something about the relationship of addition and multiplication among natural numbers. There is an analogy at the root of this idea, which is the easiest way to describe it. Prime numbers are "elementary particles" that build composite numbers by multiplying one another. At the same time, these "particles" will be ordered by addition. In the zeta function, both aspects are related to each other in the form of a sum or a product. (addition/prime numbers and multiplication/products of prime numbers)
So this is proof that the order discovered by Riemann does exist. Hence the sequence of prime numbers and their products is not similar to a random arithmetic sequence, but to a structure ordered in proportion ½. Such basic numbers are not determined by nature by the random roll of the dice. Chance and chaos are unacceptable in mathematics. I am happy to announce to everyone that one of the greatest unsolved problems in mathematics the Riemann Hypothesis, that all the so-called non-trivially zeros (unreal) have a real part equal to ½, it has just been proved by me!
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E
D
16 TABLES OF PRIMES FROM 2 TO 10,069
17
18
19
20 ∑d(p) +∑d [p(p’)]> 3 = 3(q) 4
∑[p(p’)]> 3 0
π(x) 4
30
300 000 000 000 000 000 000
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 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
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 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
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 33 333 333 334 300 000 000 000 333 333 333 334 3 000 000 000 000 3 333 333 333 334 30 000 000 000 000 33 333 333 333 334 300 000 000 000 000 333 333 333 333 334 3 000 000 000 000 000 3 333 333 333 333 334 30 000 000 000 000 000 33 333 333 333 333 334 300 000 000 000 000 000 333 333 333 333 333 334 3 000 000 000 000 000 000 3 333 333 333 333 333 334 30 000 000 000 000 000 000 33 333 333 333 333 333 334
333 333 333 333 333 333 333 334
314,897,733,565,984,132,465,468
18 435 599 767 349 200 867 866
3 000 000 000 000 000 000 000 000
2,841,589,290,368,205,431,456,186
158,410,709,631,794,568,543,814
3 333 333 333 333 333 333 333 334
3,156,487,023,934,189,563,921,654
176 846 309 399 143 769 411 680
N 10 10² 10³ 10⁴ 10⁵ 10⁶ 10⁷ 10⁸ 10⁹ 10¹⁰ 10¹¹ 10¹² 10¹³ 10¹⁴ 10¹⁵ 10¹⁶ 10¹⁷ 10¹⁸ 10¹⁹ 10²⁰ 10²¹ 10²² 10²³ 10²⁴ 10²⁵
21 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
10²⁶ 10²⁷ 10²⁸